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%I A337617 #16 Jan 09 2024 14:00:23
%S A337617 1,4,6,18,24,28,88,112,128,120,450,560,640,640,496,2364,2904,3328,
%T A337617 3456,3072,2016,12642,15400,17696,18816,17920,14336,8128,68464,82912,
%U A337617 95488,103168,102400,90112,65536,32640,374274,451296,520704,569088,580608,540672,442368,294912,130816
%N A337617 T(n, k) = (n + 1)*2^(n + k)*hypergeom([-n, k - n + 1], [2], 1/2), triangle read by rows for 0 <= k <= n.
%F A337617 T(n, k) = if n = k then 2^n*(2^(n+1)-1), otherwise 2^(2*k+1)*Sum_{j=0..n-k} ((-1)^j*2^(n-k-j)*binomial(n+1,j)*binomial(2*n-j-k,n)). - _Detlef Meya_, Dec 20 2023
%e A337617 Triangle starts:
%e A337617 [0]                                1
%e A337617 [1]                              4, 6
%e A337617 [2]                           18, 24, 28
%e A337617 [3]                       88, 112, 128, 120
%e A337617 [4]                    450, 560, 640, 640, 496
%e A337617 [5]               2364, 2904, 3328, 3456, 3072, 2016
%e A337617 [6]         12642, 15400, 17696, 18816, 17920, 14336, 8128
%e A337617 [7]    68464, 82912, 95488, 103168, 102400, 90112, 65536, 32640
%p A337617 T := (n, k) -> simplify((n + 1)*2^(n + k)*hypergeom([-n, k - n + 1], [2], 1/2)): seq(seq(T(n, k), k=0..n), n=0..8);
%t A337617 T[n_,k_] := If[n==k, 2^n*(2^(n+1)-1), 2^(2*k+1)*Sum[(-1)^j*2^(n-k-j)* Binomial[n+1, j]*Binomial[2*n-j-k, n], {j, 0, n-k}]];
%t A337617 Flatten[Table[T[n,k], {n,0,10}, {k,0,n}]] (* _Detlef Meya_, Dec 20 2023 *)
%Y A337617 T(n, n) = A171476(n) = A006516(n+1). T(n, 0) = A050146(n+1).
%Y A337617 Cf. A337992 (row sums).
%K A337617 nonn,tabl
%O A337617 0,2
%A A337617 _Peter Luschny_, Oct 19 2020

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