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%I A078812 #156 Feb 18 2025 08:58:25
%S A078812 1,2,1,3,4,1,4,10,6,1,5,20,21,8,1,6,35,56,36,10,1,7,56,126,120,55,12,
%T A078812 1,8,84,252,330,220,78,14,1,9,120,462,792,715,364,105,16,1,10,165,792,
%U A078812 1716,2002,1365,560,136,18,1,11,220,1287,3432,5005,4368,2380,816,171,20,1
%N A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
%C A078812 Warning: formulas and programs sometimes refer to offset 0 and sometimes to offset 1.
%C A078812 Apart from signs, identical to A053122.
%C A078812 Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. - _Philippe Deléham_, Feb 16 2004
%C A078812 T(n,k) is the number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2) = 10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - _Emeric Deutsch_, Apr 09 2005
%C A078812 T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - _Abdullahi Umar_, Oct 02 2008
%C A078812 This sequence is jointly generated with A085478 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Feb 25 2012
%C A078812 Concerning Kimberling's recursion relations, see A102426. - _Tom Copeland_, Jan 19 2016
%C A078812 Subtriangle of the triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 27 2012
%C A078812 From _Wolfdieter Lang_, Aug 30 2012: (Start)
%C A078812 With offset [0,0] the triangle with entries R(n,k) = T(n+1,k+1):= binomial(n+k+1, 2*k+1), n >= k >= 0, and zero otherwise, becomes the Riordan lower triangular convolution matrix R = (G(x)/x, G(x)) with G(x):=x/(1-x)^2 (o.g.f. of A000027). This means that the o.g.f. of column number k of R is (G(x)^(k+1))/x. This matrix R is the inverse of the signed Riordan lower triangular matrix A039598, called in a comment there S.
%C A078812 The Riordan matrix with entries R(n,k), just defined, provides the transition matrix between the sequence entry F(4*m*(n+1))/L(2*l), with m >= 0, for n=0,1,... and the sequence entries 5^k*F(2*m)^(2*k+1) for k = 0,1,...,n, with F=A000045 (Fibonacci) and L=A000032 (Lucas). Proof: from the inverse of the signed triangle Riordan matrix S used in a comment on A039598.
%C A078812 For the transition matrix R (T with offset [0,0]) defined above, row n=2: F(12*m) /L(2*m) = 3*5^0*F(2*m)^1 + 4*5^1*F(2*m)^3 + 1*5^2*F(2*m)^5, m >= 0. (End)
%C A078812 From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - _Tom Copeland_, Oct 11 2014
%C A078812 For 1 <= k <= n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01. - _Milan Janjic_, Dec 20 2016
%C A078812 The infinite sum (Sum_{i >= 0} (T(s+i,1+i) / 2^(s+2*i)) * zeta(s+1+2*i)) = 1 allows any zeta(s+1) to be expressed as a sum of rational multiples of zeta(s+1+2*i) having higher arguments. For example, zeta(3) can be expressed as a sum involving zeta(5), zeta(7), etc. The summation for each s >= 1 uses the s-th diagonal of the triangle. - _Robert B Fowler_, Feb 23 2022
%C A078812 The convolution triangle of the nonnegative integers. - _Peter Luschny_, Oct 07 2022
%H A078812 T. D. Noe, <a href="/A078812/b078812.txt">Rows n = 0..50 of triangle, flattened</a>
%H A078812 J. P. Allouche and M. Mendes France, <a href="http://arxiv.org/abs/1202.0211">Stern-Brocot polynomials and power series</a>, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
%H A078812 Peter Bala, <a href="/A033184/a033184.pdf">Factorisations of some Riordan arrays as infinite products</a>
%H A078812 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6.
%H A078812 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A078812 Paul Barry, <a href="https://arxiv.org/abs/2011.10827">Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers</a>, arXiv:2011.10827 [math.CO], 2020.
%H A078812 P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
%H A078812 Nour-Eddine Fahssi, <a href="https://arxiv.org/abs/1808.00045">On the combinatorics of exclusion in Haldane fractional statistics</a>, arXiv:1808.00045 [cond-mat.stat-mech], 2018.
%H A078812 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%H A078812 A. Laradji, and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-005-0553-6">Combinatorial results for semigroups of order-preserving full transformations</a>, Semigroup Forum 72 (2006), 51-62.
%H A078812 Yidong Sun, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical triangles and several classical sequences</a>, Fib. Quart. 43, no. 4, (2005) 359-370.
%F A078812 G.f.: x*y / (1 - (2 + y)*x + x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.
%F A078812 From _Philippe Deléham_, Feb 16 2004: (Start)
%F A078812 If indexing begins at 0 we have
%F A078812 T(n,k) = (n+k+1)!/((n-k)!*(2k+1))!.
%F A078812 T(n,k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n < k.
%F A078812 T(n,k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n,k) = 0 if k < 0, T(0, 0)=1 and T(0, k) = 0 for k > 0.
%F A078812 G.f. for the column k: Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2).
%F A078812 Row sums: Sum_{k>=0} T(n, k) = A001906(n+1). (End)
%F A078812 Antidiagonal sums are A000079(n) = Sum_{k=0..floor(n/2)} binomial(n+k+1, n-k). - _Paul Barry_, Jun 21 2004
%F A078812 Riordan array (1/(1-x)^2, x/(1-x)^2). - _Paul Barry_, Oct 22 2006
%F A078812 T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + 2*T(n-1,k) - T(n-2,k). - _Philippe Deléham_, Jan 26 2010
%F A078812 For another version see A128908. - _Philippe Deléham_, Mar 27 2012
%F A078812 T(n,m) = Sum_{k=0..n-m} (binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)* binomial(n+1,m+k+1)). - _Vladimir Kruchinin_, Apr 13 2016
%F A078812 T(n, k) = T(n-1, k) + (T(n-1, k-1) + T(n-2, k-1) + T(n-3, k-1) + ...) for k >= 2 with T(n, 1) = n. - _Peter Bala_, Feb 11 2025
%e A078812 Triangle begins, 1 <= k <= n:
%e A078812 1
%e A078812 2 1
%e A078812 3 4 1
%e A078812 4 10 6 1
%e A078812 5 20 21 8 1
%e A078812 6 35 56 36 10 1
%e A078812 7 56 126 120 55 12 1
%e A078812 8 84 252 330 220 78 14 1
%e A078812 From _Peter Bala_, Feb 11 2025: (Start)
%e A078812 The array factorizes as an infinite product of lower triangular arrays:
%e A078812 / 1 \ / 1 \ / 1 \ / 1 \
%e A078812 | 2 1 | | 2 1 | | 0 1 | | 0 1 |
%e A078812 | 3 4 1 | = | 3 2 1 | | 0 2 1 | | 0 0 1 | ...
%e A078812 | 4 10 6 1 | | 4 3 2 1 | | 0 3 2 1 | | 0 0 2 1 |
%e A078812 | 5 20 21 8 1| | 5 4 3 2 1| | 0 4 3 2 1 | | 0 0 3 2 1 |
%e A078812 |... | |... | |... | |... |
%e A078812 Cf. A092276. (End)
%p A078812 for n from 1 to 11 do seq(binomial(n+k-1,2*k-1),k=1..n) od; # yields sequence in triangular form; _Emeric Deutsch_, Apr 09 2005
%p A078812 # Uses function PMatrix from A357368. Adds a row and column above and to the left.
%p A078812 PMatrix(10, n -> n); # _Peter Luschny_, Oct 07 2022
%t A078812 (* First program *)
%t A078812 u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
%t A078812 u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
%t A078812 v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
%t A078812 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A078812 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A078812 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A078812 TableForm[cu]
%t A078812 Flatten[%] (* A085478 *)
%t A078812 Table[Expand[v[n, x]], {n, 1, z}]
%t A078812 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A078812 TableForm[cv]
%t A078812 Flatten[%] (* A078812 *) (* _Clark Kimberling_, Feb 25 2012 *)
%t A078812 (* Second program *)
%t A078812 Table[Binomial[n+k+1, 2*k+1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 01 2019 *)
%o A078812 (PARI) {T(n, k) = if( n<0, 0, binomial(n+k-1, 2*k-1))};
%o A078812 (PARI) {T(n, k) = polcoeff( polcoeff( x*y / (1 - (2 + y) * x + x^2) + x * O(x^n), n), k)};
%o A078812 (Haskell)
%o A078812 a078812 n k = a078812_tabl !! n !! k
%o A078812 a078812_row n = a078812_tabl !! n
%o A078812 a078812_tabl = [1] : [2, 1] : f [1] [2, 1] where
%o A078812 f us vs = ws : f vs ws where
%o A078812 ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
%o A078812 (us ++ [0, 0])
%o A078812 -- _Reinhard Zumkeller_, Dec 16 2013
%o A078812 (Sage)
%o A078812 @cached_function
%o A078812 def T(k,n):
%o A078812 if k==n: return 1
%o A078812 if k==0: return 0
%o A078812 return sum(i*T(k-1,n-i) for i in (1..n-k+1))
%o A078812 A078812 = lambda n,k: T(k,n)
%o A078812 [[A078812(n,k) for k in (1..n)] for n in (1..8)] # _Peter Luschny_, Mar 12 2016
%o A078812 (Sage) [[binomial(n+k+1, 2*k+1) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 01 2019
%o A078812 (Maxima)
%o A078812 T(n,m):=sum(binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)*binomial(n+1,m+k+1),k,0,n-m); /* _Vladimir Kruchinin_, Apr 13 2016 */
%o A078812 (Magma) /* As triangle */ [[Binomial(n+k-1, 2*k-1): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Jun 01 2018
%o A078812 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k+1, 2*k+1) ))); # _G. C. Greubel_, Aug 01 2019
%Y A078812 This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.
%Y A078812 Row sums give A001906. With signs: A053122.
%Y A078812 The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.
%Y A078812 Cf. A128908, A053123, A049310, A119900, A085478, A029653, A106195.
%K A078812 easy,nice,nonn,tabl
%O A078812 0,2
%A A078812 _Michael Somos_, Dec 05 2002
%E A078812 Edited by _N. J. A. Sloane_, Apr 28 2008
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