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Search: a006043 -id:a006043
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A027472
Third convolution of the powers of 3 (A000244).
+10
43
1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070, 805447196631, 2646469360359, 8661172452084, 28242953648100, 91789599356325, 297398301914493, 960825283108362, 3095992578904722
(list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
COMMENTS
Third column of A027465.
With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's. - Zerinvary Lajos, Dec 29 2007
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).
FORMULA
Numerators of sequence a[3,n] in (b^2)[i,j]) where b[i,j] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
From Wolfdieter Lang: (Start)
a(n) = 3^(n-3)*binomial(n-1, 2).
G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)
a(n) = |A075513(n, 2)|/9, n >= 3.
a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - Paul Curtz, Jan 07 2009
The sequence 0, 1, 9, 54, ... has e.g.f.: (x + 3*x^2/2)*exp(3*x)/. - Paul Barry, Jul 23 2003
E.g.f.: E(0) where E(k) = 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
With offset=2 e.g.f.: x^2*exp(3*x)/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).
Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)
MATHEMATICA
nn=41; Drop[Range[0, nn]!CoefficientList[Series[Exp[x]^3 x^2/2!, {x, 0, nn}], x], 2] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{9, -27, 27}, {1, 9, 54}, 40] (* G. C. Greubel, May 12 2021 *)
Abs[Take[CoefficientList[Series[1/(1+3x^2)^3, {x, 0, 60}], x], {1, -1, 2}]] (* Harvey P. Dale, Mar 03 2022 *)
PROG
(Sage) [3^(n-3)*binomial(n-1, 2) for n in range(3, 40)] # Zerinvary Lajos, Mar 10 2009
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 27, -27, 9]^(n-3)*[1; 9; 54])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(Magma) [3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // G. C. Greubel, May 12 2021
CROSSREFS
Cf. A000244, A006043, A027465, A075513, A152818.
Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).
KEYWORD
nonn,easy
AUTHOR
Olivier Gérard
EXTENSIONS
Corrected by T. D. Noe, Nov 07 2006
Better name from Wolfdieter Lang
Terms a(23) onward added by G. C. Greubel, May 12 2021
STATUS
approved
A152818
Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.
+10
14
1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.
Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009
LINKS
G. C. Greubel, Antidiagonals n = 0..50, flattened
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.
FORMULA
E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From Peter Bala, Oct 09 2011: (Start)
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.
Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)
From G. C. Greubel, Apr 10 2023: (Start)
T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)
EXAMPLE
From Omar E. Pol, Jan 06 2009: (Start)
Array begins:
1, 1, 2, 6, 24, 120, ...
1, 4, 18, 96, 600, 4320, ...
1, 12, 108, 960, 9000, 90720, ...
1, 32, 540, 7680, 105000, 1451520, ...
1, 80, 2430, 53760, 1050000, 19595520, ...
1, 192, 10206, 344064, 9450000, 235146240, ...
1, 448, 40824, 2064384, 78750000, 2586608640, ...
1, 1024, 157464, 11796480, 618750000, 26605117440, ...
1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
1;
1, 1;
1, 4, 2;
1, 12, 18, 6;
1, 32, 108, 96, 24;
1, 80, 540, 960, 600, 120;
1, 192, 2430, 7680, 9000, 4320, 720;
1, 448, 10206, 53760, 105000, 90720, 35280, 5040;
MATHEMATICA
len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n, 0, m}, {k, 0, m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)
PROG
(Sage)
def A152818_row(n):
R.<x> = ZZ[]
P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
return P.coefficients()
for n in (0..12): print(A152818_row(n)) # Peter Luschny, May 03 2013
(PARI) A(n, k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016
(Magma)
A152818:= func< n, k | (k+1)^(n-k)*Factorial(k)*Binomial(n, k) >;
[A152818(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023
CROSSREFS
Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.
Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.
Cf. A152656, A154372.
KEYWORD
nonn,tabl
AUTHOR
Paul Curtz, Dec 13 2008
EXTENSIONS
Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009
STATUS
approved
A006044
a(n) = 4^(n-4)*(n-1)*(n-2)*(n-3).
(Formerly M4290)
+10
6
6, 96, 960, 7680, 53760, 344064, 2064384, 11796480, 64880640, 346030080, 1799356416, 9160359936, 45801799680, 225485783040, 1095216660480, 5257039970304, 24970939858944, 117510305218560, 548381424353280, 2539871860162560, 11683410556747776, 53409876830846976
(list; graph; refs; listen; history; text; internal format)
OFFSET
4,1
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
Frank A. Haight, Letter to N. J. A. Sloane, n.d..
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).
FORMULA
G.f. = 6*x^4/(1-4*x)^4. - Emeric Deutsch, Apr 29 2004
a(n) = 6*A038846(n). - R. J. Mathar , Mar 22 2013
E.g.f.: (3 + exp(4*x)*(32*x^3 - 24*x^2 + 12*x - 3))/128. - Stefano Spezia, Jan 01 2023
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=4} 1/a(n) = 18*log(4/3) - 5.
Sum_{n>=4} (-1)^n/a(n) = 50*log(5/4) - 11. (End)
MATHEMATICA
a[n_] := 4^(n - 4)*(n - 1)*(n - 2)*(n - 3); Array[a, 25, 4] (* Amiram Eldar, Jan 08 2023 *)
PROG
(Magma) [4^(n-4)*(n-3)*(n-2)*(n-1): n in [4..30]]; // Vincenzo Librandi, Aug 14 2011
CROSSREFS
Cf. A000142, A006043, A038846, A154120.
Column k=3 of square array A152818. - Paul Curtz, Dec 17 2008 [corrected by Omar E. Pol, Jan 07 2009]
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Emeric Deutsch, Apr 29 2004
Erroneous reference deleted by Martin J. Erickson (erickson(AT)truman.edu), Nov 03 2010
Entry revised by N. J. A. Sloane, Dec 27 2021
STATUS
approved
A154372
Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).
+10
4
1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
1;
-1, 1;
3, -4, 1;
-16, 24, -9, 1;
125, -200, 90, -16, 1;
-1296, 2160, -1080, 240, -25, 1;
16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
FORMULA
T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)
EXAMPLE
With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1 \ /1 \ /1 \ /1 \
|1 1 ||0 1 ||0 1 | |1 1 |
|1 3 1 ||0 1 1 ||0 0 1 |... = |1 4 1 |
|1 6 5 1 ||0 1 3 1 ||0 0 1 1 | |1 12 9 1|
|... ||0 1 6 5 1 ||0 0 1 3 1| |... |
|... ||... ||... | | |
- Peter Bala, Jan 13 2015
MATHEMATICA
T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)
PROG
(Magma) /* As triangle */ [[(k+1)^(n-k)*Binomial(n, k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016
CROSSREFS
Cf. A080108, A152818, A059297, A145033.
KEYWORD
nonn,easy,tabl
AUTHOR
Paul Curtz, Jan 08 2009
STATUS
approved
A154128
a(n) = 5^n*(n+4)!/n!.
+10
1
24, 600, 9000, 105000, 1050000, 9450000, 78750000, 618750000, 4640625000, 33515625000, 234609375000, 1599609375000, 10664062500000, 69726562500000, 448242187500000, 2838867187500000, 17742919921875000, 109588623046875000
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Column 4 of square array A152818.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (25, -250, 1250, -3125, 3125).
FORMULA
a(n) = 5^n*(n+4)*(n+3)*(n+2)*(n+1).
From R. J. Mathar, Feb 06 2009: (Start)
a(n) = A052762(n+4)*A000351(n).
a(n) = 24*A036071(n).
G.f: 24/(1-5*x)^5. (End)
From G. C. Greubel, Sep 02 2016: (Start)
a(n) = 25*a(n-1) - 250*a(n-2) + 1250*a(n-3) - 3125*a(n-4) + 3125*a(n-5).
E.g.f.: (24 + 480*x + 1800*x^2 + 2000*x^3 + 625*x^4)*exp(5*x). (End)
MATHEMATICA
LinearRecurrence[{25, -250, 1250, -3125, 3125}, {24, 600, 9000, 105000, 1050000}, 25] (* or *) Table[5^n*(n+4)*(n+3)*(n+2)*(n+1), {n, 0, 25}] (* G. C. Greubel, Sep 02 2016 *)
PROG
(Magma) [5^n*(n+4)*(n+3)*(n+2)*(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011
CROSSREFS
Cf. A000142, A006043, A006044, A152818, A154120.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jan 05 2009
EXTENSIONS
More terms from R. J. Mathar, Feb 06 2009
STATUS
approved
page 1

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