Displaying 1-10 of 23 results found.
Powers of 6: a(n) = 6^n.
(Formerly M4224 N1765)
+10
177
1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444736, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856, 131621703842267136
COMMENTS
Same as Pisot sequences E(1, 6), L(1, 6), P(1, 6), T(1, 6). Essentially same as Pisot sequences E(6, 36), L(6, 36), P(6, 36), T(6, 36). See A008776 for definitions of Pisot sequences. Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.
a(n) is number of compositions of natural numbers into n parts less than 6. For example, a(2) = 36, and there are 36 compositions of natural numbers into 2 parts less than 6.
The compositions of n in which each part is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color.
Number of words of length n over the alphabet of six letters. - Joerg Arndt, Sep 16 2014 The number of ordered triples (x, y, z) of binary words of length n such that D(x,z) = D(x, y) + D(y, z) where D(a, b) is the Hamming distance from a to b. - Geoffrey Critzer, Mar 06 2017 a(n) is the area of a triangle with vertices at (2^n, 3^n), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^(n+2)); a(n) is also one fifth the area of a triangle with vertices at (2^n, 3^(n+2)), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^n). - J. M. Bergot, May 07 2018 a(n) is the number of possible outcomes of n distinguishable 6-sided dice. - Stefano Spezia, Jul 06 2024
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = 6^n.
a(0) = 1; a(n) = 6*a(n-1).
E.g.f.: exp(6*x).
a(n) = det(|s(i+3,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013
PROG
(Haskell)
a000400 = (6 ^)
(Scala) (List.fill(50)(6: BigInt)).scanLeft(1: BigInt)(_ * _) // Alonso del Arte, May 31 2019
a(n) = Sum_{k=0..n} (n-k+1)^k.
+10
46
1, 2, 4, 9, 23, 66, 210, 733, 2781, 11378, 49864, 232769, 1151915, 6018786, 33087206, 190780213, 1150653921, 7241710930, 47454745804, 323154696185, 2282779990495, 16700904488706, 126356632390298, 987303454928973, 7957133905608837, 66071772829247410
COMMENTS
Antidiagonal sums of array A003992. a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015 Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016 Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:
{{1},{2},{3},{4}}
{{1},{2},{3,4}}
{{1},{2,4},{3}}
{{1,4},{2},{3}}
{{1},{2,3,4}}
{{1,3},{2,4}}
{{1,4},{2,3}}
{{1,3,4},{2}}
{{1,2,3,4}}
Missing from this list are:
{{1},{2,3},{4}}
{{1,2},{3},{4}}
{{1,3},{2},{4}}
{{1,2},{3,4}}
{{1,2,3},{4}}
{{1,2,4},{3}}
(End)
a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021
FORMULA
G.f.: G(0) where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013 E.g.f.: Sum_{n>=0} Integral^n exp((n+1)*x) dx^n, where Integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013 O.g.f.: Sum_{n>=0} n! * x^n/(1-x)^(n+1) / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Jul 20 2014 a(n-1) = Sum_{k = 1..n} k^(n-k). - Gus Wiseman, Jan 08 2019 log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021 a(n) ~ sqrt(2*Pi/(n+1 + (n+1)/w(n))) * ((n+1)/w(n))^(n+2 - (n+1)/w(n)), where w(n) = LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 66*x^5 + 210*x^6 + ...
where we have the identity:
A(x) = 1/(1-x) + x/(1-2*x) + x^2/(1-3*x) + x^3/(1-4*x) + x^4/(1-5*x) + ...
is equal to
A(x) = 1/(1-x) + x/((1-x)^2*(1+x)) + 2!*x^2/((1-x)^3*(1+x)*(1+2*x)) + 3!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^4/((1-x)^5*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
The a(5-1) = 23 RGS described in the comment are (dots denote zeros):
01: [ . . . . . ]
02: [ . 1 . . . ]
03: [ . 1 . . 1 ]
04: [ . 1 . 1 . ]
05: [ . 1 . 1 1 ]
06: [ . 1 1 . . ]
07: [ . 1 1 . 1 ]
08: [ . 1 1 1 . ]
09: [ . 1 1 1 1 ]
10: [ . 1 2 . . ]
11: [ . 1 2 . 1 ]
12: [ . 1 2 . 2 ]
13: [ . 1 2 1 . ]
14: [ . 1 2 1 1 ]
15: [ . 1 2 1 2 ]
16: [ . 1 2 2 . ]
17: [ . 1 2 2 1 ]
18: [ . 1 2 2 2 ]
19: [ . 1 2 3 . ]
20: [ . 1 2 3 1 ]
21: [ . 1 2 3 2 ]
22: [ . 1 2 3 3 ]
23: [ . 1 2 3 4 ]
(End)
MAPLE
a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/(1-(m+1)*x+x*O(x^n))), n)} /* Paul D. Hanna, Sep 13 2011 */ (PARI) {INTEGRATE(n, F)=local(G=F); for(i=1, n, G=intformal(G)); G}
{a(n)=local(A=1+x); A=sum(k=0, n, INTEGRATE(k, exp((k+1)*x+x*O(x^n)))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Dec 28 2013 for(n=0, 30, print1(a(n), ", "))
(PARI)
{a(n)=polcoeff( sum(m=0, n, m!*x^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x +x*O(x^n))), n)} /* From o.g.f. ( Paul D. Hanna, Jul 20 2014) */ for(n=0, 25, print1(a(n), ", "))
(Haskell)
a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..]
(Magma) [(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019 (Sage) [sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021
Array read by descending antidiagonals: T(n,k) is the number of unoriented strings with n beads of k or fewer colors.
+10
33
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 6, 1, 0, 1, 5, 10, 18, 10, 1, 0, 1, 6, 15, 40, 45, 20, 1, 0, 1, 7, 21, 75, 136, 135, 36, 1, 0, 1, 8, 28, 126, 325, 544, 378, 72, 1, 0, 1, 9, 36, 196, 666, 1625, 2080, 1134, 136, 1, 0, 1, 10, 45, 288, 1225, 3996, 7875, 8320, 3321, 272, 1, 0
COMMENTS
Column k of this array is the "BIK" (reversible, indistinct, unlabeled) transform of k,0,0,0,....
Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BIK(c_k(n): n >= 1) be the output sequence under Bower's BIK transform. It can proved that the g.f. of BIK(c_k(n): n >= 1) is A_k(x) = (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))). (See the comments for sequence A001224.) For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k is (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))) = (1/2)*(k*x/(1-k*x) + (k*x^2 + k*x)/(1-k*x^2)) = (2 + (1-k)*x - 2*k*x^2)*k*x/(2*(1-k*x^2)*(1-k*x)).
Using the first form the g.f. above and the expansion 1/(1-y) = 1 + y + y^2 + ..., we can easily prove J.-F. Alcover's formula T(n,k) = (k^n + k^((n + mod(n,2))/2))/2.
(End)
FORMULA
T(n,k) = [n==0] + [n>0] * (k^n + k^ceiling(n/2)) / 2. [Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018] G.f. for row n: (Sum_{j=0..n} S2(n,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=0..ceiling(n/2)} S2(ceiling(n/2),j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277. G.f. for row n>0: x*Sum_{k=0..n-1} A145882(n,k) * x^k / (1-x)^(n+1). E.g.f. for row n: (Sum_{k=0..n} S2(n,k)*x^k + Sum_{k=0..ceiling(n/2)} S2(ceiling(n/2),k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277. T(0,k) = 1; T(1,k) = k; T(2,k) = binomial(k+1,2); for n>2, T(n,k) = k*(T(n-3,k)+T(n-2,k)-k*T(n-1,k)).
For k>n, T(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * T(n,k-j). (End)
EXAMPLE
Array begins with T(0,0):
1 1 1 1 1 1 1 1 1 1 ...
0 1 2 3 4 5 6 7 8 9 ...
0 1 3 6 10 15 21 28 36 45 ...
0 1 6 18 40 75 126 196 288 405 ...
0 1 10 45 136 325 666 1225 2080 3321 ...
0 1 20 135 544 1625 3996 8575 16640 29889 ...
0 1 36 378 2080 7875 23436 58996 131328 266085 ...
0 1 72 1134 8320 39375 140616 412972 1050624 2394765 ...
0 1 136 3321 32896 195625 840456 2883601 8390656 21526641 ...
0 1 272 9963 131584 978125 5042736 20185207 67125248 193739769 ...
0 1 528 29646 524800 4884375 30236976 141246028 536887296 1743421725 ...
...
MATHEMATICA
Table[If[n>0, ((n-k)^k + (n-k)^Ceiling[k/2])/2, 1], {n, 0, 15}, {k, 0, n}] // Flatten (* updated Jul 10 2018 *) (* Adapted to T(0, k)=1 by Robert A. Russell, Nov 13 2018 *)
PROG
(PARI) for(n=0, 15, for(k=0, n, print1(if(n==0, 1, ((n-k)^k + (n-k)^ceil(k/2))/2), ", "))) \\ G. C. Greubel, Nov 15 2018 (PARI) T(n, k) = {(k^n + k^ceil(n/2)) / 2} \\ Andrew Howroyd, Sep 13 2019 (Magma) [[n le 0 select 1 else ((n-k)^k + (n-k)^Ceiling(k/2))/2: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Nov 15 2018
CROSSREFS
Rows 0-20 are A000012, A001477, A000217 (triangular numbers), A002411 (pentagonal pyramidal numbers), A037270, A168178, A071232, A168194, A071231, A168372, A071236, A168627, A071235, A168663, A168664, A170779, A170780, A170790, A170791, A170801, A170802.
EXTENSIONS
Array transposed for greater consistency by Andrew Howroyd, Apr 04 2017
Array read by ascending antidiagonals: A(n, k) = k^n.
+10
22
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 27, 16, 5, 1, 0, 1, 32, 81, 64, 25, 6, 1, 0, 1, 64, 243, 256, 125, 36, 7, 1, 0, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 0, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 0, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1
COMMENTS
This array transforms into A371761 using the Akiyama-Tanigawa algorithm for powers applied to the rows. - Peter Luschny, Apr 16 2024 This array transforms into A344499 using the Akiyama-Tanigawa algorithm for powers applied to the columns. - Peter Luschny, Apr 27 2024
FORMULA
Table of x^y, where (x,y) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...
As a number triangle, columns have g.f. x^k/(1 - kx). - Paul Barry, Mar 28 2005 T(n, k) = if(k <= n, k^(n - k), 0).
T(n, k) = Sum_{j=0..floor((n-k)/2)} (-1)^j*C(n-k, j)*C(n-k-j, n-k)*k^(n-k-2j).
(End)
EXAMPLE
Seen as an array that is read by ascending antidiagonals:
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
[2] 0, 1, 4, 9, 16, 25, 36, 49, 64, ...
[3] 0, 1, 8, 27, 64, 125, 216, 343, 512, ...
[4] 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, ...
[5] 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, ...
[6] 0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, ...
[7] 0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, ...
MATHEMATICA
T[x_, y_] := If[y == 0, 1, (x - y)^y];
PROG
(PARI)
(SageMath)
def Arow(n, len): return [k**n for k in range(len)]
for n in range(8): print([n], Arow(n, 9)) # Peter Luschny, Apr 16 2024
Table T(n,k) = k^n read by upwards antidiagonals (n >= 1, k >= 1).
+10
18
1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 16, 27, 16, 5, 1, 32, 81, 64, 25, 6, 1, 64, 243, 256, 125, 36, 7, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1, 1024, 19683, 65536, 78125, 46656, 16807, 4096, 729, 100, 11
FORMULA
a(n) = (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1), where b(n) = [ 1/2 + sqrt(2 * n) ]. (b(n) is the n-th term of A002024.) - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002
EXAMPLE
1 2 3 4 5 6 7
1 4 9 16 25 36 49
1 8 27 64 125 216 343
1 16 81 256 625 1296 2401
1 32 243 1024 3125 7776 16807
1 64 729 4096 15625 46656 117649
1 128 2187 16384 78125 279936 823543
PROG
(Haskell)
a051129 n k = k ^ (n - k)
a051129_row n = a051129_tabl !! (n-1)
a051129_tabl = zipWith (zipWith (^)) a002260_tabl $ map reverse a002260_tabl
(PARI) b(n) = floor(1/2 + sqrt(2 * n));
vector(100, n, (n - b(n) * (b(n) - 1) / 2)^(b(n) * (b(n) + 1) / 2 - n + 1)) \\ Altug Alkan, Dec 09 2015
Table T(n,k) = n^k read by upwards antidiagonals (n >= 1, k >= 1).
+10
12
1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 11, 100, 729, 4096, 16807, 46656, 78125, 65536, 19683, 1024, 1
EXAMPLE
Table begins
1, 1, 1, 1, 1, ...
2, 4, 8, 16, 32, ...
3, 9, 27, 81, 243, ...
4, 16, 64, 256, 1024, ...
MAPLE
a := floor((sqrt(8*n-7)+1)/2);
b := (a+a^2)/2-n;
c := (a-a^2)/2+n;
(b+1)^c end:
# second Maple program:
T:= (n, k)-> n^k:
a(n) = max_{k=0..n} k^(n-k).
(Formerly M1198)
+10
11
1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, 4096, 16384, 78125, 390625, 1953125, 10077696, 60466176, 362797056, 2176782336, 13841287201, 96889010407, 678223072849, 4747561509943, 35184372088832, 281474976710656, 2251799813685248
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 231.
EXAMPLE
a(5) = max(5^0, 4^1, 3^2, 2^3, 1^4, 0^5) = max(1,4,9,8,1,0) = 9.
MATHEMATICA
Join[{1}, Max[#]&/@Table[k^(n-k), {n, 25}, {k, n}]] (* Harvey P. Dale, Jun 20 2011 *)
PROG
(Haskell)
a003320 n = maximum $ zipWith (^) [0 .. n] [n, n-1 ..]
(PARI) a(n) = vecmax(vector(n+1, k, (k-1)^(n-k+1))); \\ Michel Marcus, Jun 13 2017
EXTENSIONS
Easdown reference from Michail Kats (KatsMM(AT)info.sgu.ru)
Number of orientable strings of length n using a maximum of k colors, array read by descending antidiagonals, T(n,k) for n >= 1 and k >= 1.
+10
11
0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 0, 6, 9, 6, 0, 0, 10, 24, 36, 12, 0, 0, 15, 50, 120, 108, 28, 0, 0, 21, 90, 300, 480, 351, 56, 0, 0, 28, 147, 630, 1500, 2016, 1053, 120, 0, 0, 36, 224, 1176, 3780, 7750, 8064, 3240, 240, 0, 0, 45, 324, 2016, 8232, 23220, 38750, 32640, 9720, 496, 0
COMMENTS
Reversing the string does not leave it unchanged. Only one string from each pair is counted.
Equivalently, the number of nonequivalent strings up to reversal that are not palindromes.
Except for the first term, column k is the "BHK" (reversible, identity, unlabeled) transform of k,0,0,0,... [Corrected by Petros Hadjicostas, Jul 01 2018] Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BHK(c_k(n): n >= 1) be the output sequence under Bower's BHK transform. It can be proved that the g.f. of BHK(c_k(n): n >= 1) is A_k(x) = (C_k(x)^2 - C_k(x^2))/(2*(1-C_k(x))*(1-C_k(x^2))) + C_k(x). (See the comments for sequences A032096, A032097, and A032098.) For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k (with the term C_k(x) = k*x excluded) is (C_k(x)^2 - C_k(x^2))/(2*(1-C_k(x))*(1-C_k(x^2))) = (1/2)*(k - 1)*k*x^2/((k*x^2 - 1)*(k*x - 1)), from which we can easily prove Howroyd's formula.
(End)
We give an alternative definition for the square array A(n,k) = T(n,k) with n >= 2 and k >= 0. A(n,k) is the number of inequivalent "distinguishing colorings" of the path on n vertices using at most k colors. The rows are indexed by n, the number of vertices of the path, and the columns are indexed by k, the number of permissible colors.
A vertex-coloring of a graph G is called "distinguishing" if it is only preserved by the identity automorphism of G. This notion is considered in the context of "symmetry breaking" of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called "equivalent" if there is an automorphism of the graph which preserves the colors of the vertices. Given a graph G, we use the notation Phi_k(G) to denote the number of inequivalent distinguishing colorings of G with at most k colors. This sequence gives A(n,k) = Phi_k(P_n), i.e., the number of inequivalent distinguishing colorings of the path P_n on n vertices with at most k colors.
For n=3, we can color the vertices of P_3 with at most 2 colors in 3 ways such that all the colorings distinguish the graph (i.e., no non-identity automorphism of the path P_3 preserves the coloring) and that all the three colorings are inequivalent.
We have Phi_k(P_n) = binomial(k,2)*k^(n-2) + k*Phi_k(P_(n-2)) for n >= 4; Phi_k(P_2) = binomial(k,2); Phi_k(P_3) = k*binomial(k,2).
(End)
FORMULA
T(n,k) = (k^n - k^(ceiling(n/2)))/2.
G.f. for column k: (1/2)*(k - 1)*k*x^2/((k*x^2 - 1)*(k*x - 1)). - Petros Hadjicostas, Jul 07 2018 G.f. for row n: (Sum_{j=0..n} S2(n,j)*j!*x^j/(1-x)^(j+1) - Sum_{j=0..ceiling(n/2)} S2(ceiling(n/2),j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277. G.f. for row n>1: x * Sum_{k=1..n-1} A145883(n,k) * x^k / (1-x)^(n+1). E.g.f. for row n: (Sum_{k=0..n} S2(n,k)*x^k - Sum_{k=0..ceiling(n/2)} S2(ceiling(n/2),k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277. T(0,k) = T(1,k) = 0; T(2,k) = binomial(k,2); for n>2, T(n,k) = k*(T(n-3,k)+T(n-2,k)-k*T(n-1,k)).
For k>n, T(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * T(n,k-j). (End)
EXAMPLE
Array begins:
======================================================
n\k| 1 2 3 4 5 6 7 8
---|--------------------------------------------------
1 | 0 0 0 0 0 0 0 0...
2 | 0 1 3 6 10 15 21 28...
3 | 0 2 9 24 50 90 147 224...
4 | 0 6 36 120 300 630 1176 2016...
5 | 0 12 108 480 1500 3780 8232 16128...
6 | 0 28 351 2016 7750 23220 58653 130816...
7 | 0 56 1053 8064 38750 139320 410571 1046528...
8 | 0 120 3240 32640 195000 839160 2881200 8386560...
...
For T(4,2)=6, the chiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB.
MATHEMATICA
Table[Function[n, (k^n - k^(Ceiling[n/2]))/2][m - k + 1], {m, 11}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, Oct 11 2017 *)
PROG
(PARI) T(n, k) = (k^n - k^(ceil(n/2)))/2;
Array T(n,k) = k*(k-1)^(n-1) read by ascending antidiagonals; k,n >= 1.
+10
7
1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 12, 12, 5, 0, 2, 24, 36, 20, 6, 0, 2, 48, 108, 80, 30, 7, 0, 2, 96, 324, 320, 150, 42, 8, 0, 2, 192, 972, 1280, 750, 252, 56, 9, 0, 2, 384, 2916, 5120, 3750, 1512, 392, 72, 10, 0, 2, 768, 8748, 20480, 18750, 9072, 2744, 576, 90, 11
COMMENTS
T(n,k) is the number of n-letter words in a k-letter alphabet with no adjacent letters the same. The factor k represents the number of choices of the first letter, and the n-1 times repeated factor k-1 represents the choices of the next n-1 letters avoiding their predecessor.
The antidiagonal sums are s(d) = 1, 2, 5, 12, 31, 88, 275, 942, 3513, 14158, 61241, 282632, .. for d = n+k >= 2.
FORMULA
G.f. for column k: k*x/(1-(k-1)*x). - R. J. Mathar, Dec 12 2015 G.f. for array: y/(y-1) - (1+1/x)*y*LerchPhi(y,1,-1/x). - Robert Israel, Dec 13 2018
EXAMPLE
1 2 3 4 5 6 7
0 2 6 12 20 30 42
0 2 12 36 80 150 252
0 2 24 108 320 750 1512
0 2 48 324 1280 3750 9072
0 2 96 972 5120 18750 54432
0 2 192 2916 20480 93750 326592
T(3,3)=12 counts aba, abc, aca, acb, bab, bac, bca, bcb, cab, cac, cba, cbc. Words like aab or cbb are not counted.
MAPLE
k*(k-1)^(n-1) ;
end proc:
seq(seq( A265583(d-k, k), k=1..d-1), d=2..13) ;
MATHEMATICA
T[1, 1] = 1; T[n_, k_] := If[k==1, 0, k*(k-1)^(n-1)]; Table[T[n-k, k], {n, 2, 12}, {k, 1, n-1}] // Flatten (* Amiram Eldar, Dec 13 2018 *)
PROG
(PARI) T(n, k) = if(n==k==1, 1, k*(k-1)^(n-k-1) );
for(n=2, 15, for(k=1, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 10 2019 (Magma)
T:= func< n, k | (n eq 1 and k eq 1) select 1 else k*(k-1)^(n-k-1) >;
[T(n, k): k in [1..n-1], n in [2..15]]; // G. C. Greubel, Aug 10 2019 (Sage)
def T(n, k):
if (n==k==1): return 1
else: return k*(k-1)^(n-k-1)
[[T(n, k) for k in (1..n-1)] for n in (2..15)] # G. C. Greubel, Aug 10 2019 (GAP)
T:= function(n, k)
if (n=1 and k=1) then return 1;
else return k*(k-1)^(n-k-1);
fi;
end;
Flat(List([2..15], n-> List([1..n-1], k-> T(n, k) ))); # G. C. Greubel, Aug 10 2019
Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.
+10
7
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 10, 1, 0, 1, 4, 27, 56, 1, 0, 1, 5, 52, 381, 346, 1, 0, 1, 6, 85, 1192, 6219, 2252, 1, 0, 1, 7, 126, 2705, 36628, 111753, 15184, 1, 0, 1, 8, 175, 5136, 124405, 1297504, 2151549, 104960, 1, 0
COMMENTS
Let A_m(n,k) = (n!)^m [x^n] hypergeom([], [1,…,1], z)^k where [1,…,1] lists (m-1) times 1. These arrays can be seen as generalizations of the power functions n^k. For m = 1 -> A003992, m = 2 -> A287316, m = 3 -> A287698. A_m(n,n) is the sum of m-th powers of coefficients in the full expansion of (z_1+z_2+...+z_n)^n (compare A245397). A287696 provide polynomials and A287697 rational functions generating the columns of the array.
EXAMPLE
Array starts:
k\n| 0 1 2 3 4 5 6 7
---|-------------------------------------------------------------------
k=0| 1, 0, 0, 0, 0, 0, 0, 0, ... A000007k=1| 1, 1, 1, 1, 1, 1, 1, 1, ... A000012k=2| 1, 2, 10, 56, 346, 2252, 15184, 104960, ... A000172k=3| 1, 3, 27, 381, 6219, 111753, 2151549, 43497891, ... A141057k=4| 1, 4, 52, 1192, 36628, 1297504, 50419096, 2099649808, ... A287699k=5| 1, 5, 85, 2705, 124405, 7120505, 464011825, 33031599725, ...
k=6| 1, 6, 126, 5136, 316206, 25461756, 2443835736, 263581282656, ...
MAPLE
A287698_row := proc(k, len) hypergeom([], [1, 1], x):
series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
for k from 0 to 6 do A287698_row(k, 9) od; A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287698_col(n, 9) od;
MATHEMATICA
Table[Table[SeriesCoefficient[HypergeometricPFQ[{}, {1, 1}, x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)
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