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A320905 T(n, k) = binomial(2*n - 1 - k, k - 1)*hypergeom([2, 2, 1-k], [1, 1 - 2*k + 2*n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.

%I A320905 #27 Jan 01 2024 15:31:06
%S A320905 1,1,5,1,7,18,1,9,31,56,1,11,48,111,160,1,13,69,198,351,432,1,15,94,
%T A320905 325,699,1023,1120,1,17,123,500,1280,2223,2815,2816,1,19,156,731,2186,
%U A320905 4458,6562,7423,6912,1,21,193,1026,3525,8330,14198,18324,18943,16640
%N A320905 T(n, k) = binomial(2*n - 1 - k, k - 1)*hypergeom([2, 2, 1-k], [1, 1 - 2*k + 2*n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.
%H A320905 Andrew Howroyd, <a href="/A320905/b320905.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A320905 T(n, k) = Sum_{j=0..2*n-k} binomial(2*n-k, 2*n - 2*k + 1 + j)*binomial(j+2, 2). - _Detlef Meya_, Dec 31 2023
%e A320905 Triangle starts:
%e A320905 [1] 1
%e A320905 [2] 1,  5
%e A320905 [3] 1,  7,  18
%e A320905 [4] 1,  9,  31,  56
%e A320905 [5] 1, 11,  48, 111,  160
%e A320905 [6] 1, 13,  69, 198,  351,  432
%e A320905 [7] 1, 15,  94, 325,  699, 1023, 1120
%e A320905 [8] 1, 17, 123, 500, 1280, 2223, 2815, 2816
%e A320905 [9] 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912
%p A320905 T := (n, k) -> binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1):
%p A320905 seq(seq(simplify(T(n, k)), k=1..n), n=1..10);
%t A320905 T[n_, k_] := Sum[Binomial[2*n-k, 2*n-2*k+1+j]*Binomial[j+2, 2],{j, 0, 2*n-k}]; Flatten[Table[T[n, k], {n, 1, 10}, {k, 1, n}]] (* _Detlef Meya_, Dec 31 2023 *)
%o A320905 (PARI) T(n, k) = {sum(j=0, 2*n-k, binomial(2*n-k, 2*n - 2*k + 1 + j) * binomial(j+2, 2))} \\ _Andrew Howroyd_, Dec 31 2023
%o A320905 (Python)
%o A320905 from functools import cache
%o A320905 @cache
%o A320905 def T(n, k):
%o A320905     if k < 1 or n < 1: return 0
%o A320905     if k == 1: return 1
%o A320905     if k == n: return n * (n + 3) * 2**(n - 3)
%o A320905     return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
%o A320905 for n in range(1, 10): print([T(n, k) for k in range(1, n+1)])
%o A320905 # after _Detlef Meya_, _Peter Luschny_, Jan 01 2024
%Y A320905 Row sums with shifted indices in A318947.
%Y A320905 T(n, n) = A001793(n).
%K A320905 nonn,tabl
%O A320905 1,3
%A A320905 _Peter Luschny_, Oct 28 2018