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%I #24 Jan 01 2024 15:30:59
%S 0,0,1,0,1,6,0,1,8,24,0,1,10,39,80,0,1,12,58,150,240,0,1,14,81,256,
%T 501,672,0,1,16,108,406,955,1524,1792,0,1,18,139,608,1686,3178,4339,
%U 4608,0,1,20,174,870,2794,6144,9740,11762,11520
%N T(n, k) = binomial(2*n - k, k - 1)*hypergeom([2, 2, 1 - k], [1, 2*(1 - k + n)], -1), triangle read by rows, T(n,k) for n >= 0 and 0 <= k <= n.
%H Andrew Howroyd, <a href="/A320906/b320906.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F T(n, k) = Sum_{j=0..2*n+1-k} binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2). - _Detlef Meya_, Dec 31 2023
%e Triangle starts:
%e [0] 0
%e [1] 0, 1
%e [2] 0, 1, 6
%e [3] 0, 1, 8, 24
%e [4] 0, 1, 10, 39, 80
%e [5] 0, 1, 12, 58, 150, 240
%e [6] 0, 1, 14, 81, 256, 501, 672
%e [7] 0, 1, 16, 108, 406, 955, 1524, 1792
%e [8] 0, 1, 18, 139, 608, 1686, 3178, 4339, 4608
%e [9] 0, 1, 20, 174, 870, 2794, 6144, 9740, 11762, 11520
%p T := (n, k) -> binomial(2*n-k, k-1)*hypergeom([2, 2, 1-k], [1, 2*(1-k+n)], -1):
%p seq(seq(simplify(T(n, k)), k=0..n), n=0..9);
%t T[n_, k_] := Sum[Binomial[2*n+1-k, 2*n+2-2*k+j]*Binomial[j+2, 2], {j,0, 2*n+1-k}]; Flatten[Table[T[n, k], {n, 0, 15}, {k, 0, n}]] (* _Detlef Meya_, Dec 31 2023 *)
%o (PARI) T(n, k) = {sum(j=0, 2*n+1-k, binomial(2*n+1-k, 2*n+2-2*k+j) * binomial(j+2,2))} \\ _Andrew Howroyd_, Dec 31 2023
%o (Python)
%o from functools import cache
%o @cache
%o def T(n, k):
%o if k <= 0 or n <= 0: return 0
%o if k == 1: return 1
%o if k == n: return n * (n + 1) * 2**(n - 2)
%o return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
%o for n in range(10): print([T(n, k) for k in range(n + 1)])
%o # after _Detlef Meya_, _Peter Luschny_, Jan 01 2024
%Y Row sums are A320907. T(n, n) = A001788(n).
%Y Cf. A320905.
%K nonn,tabl
%O 0,6
%A _Peter Luschny_, Oct 28 2018
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