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.

1, 4, 6, 18, 24, 28, 88, 112, 128, 120, 450, 560, 640, 640, 496, 2364, 2904, 3328, 3456, 3072, 2016, 12642, 15400, 17696, 18816, 17920, 14336, 8128, 68464, 82912, 95488, 103168, 102400, 90112, 65536, 32640, 374274, 451296, 520704, 569088, 580608, 540672, 442368, 294912, 130816

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

Triangle starts:
[0]                                1
[1]                              4, 6
[2]                           18, 24, 28
[3]                       88, 112, 128, 120
[4]                    450, 560, 640, 640, 496
[5]               2364, 2904, 3328, 3456, 3072, 2016
[6]         12642, 15400, 17696, 18816, 17920, 14336, 8128
[7]    68464, 82912, 95488, 103168, 102400, 90112, 65536, 32640

Maple

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);

Mathematica

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}]];
Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Dec 20 2023 *)

Crossrefs

T(n, n) = A171476(n) = A006516(n+1). T(n, 0) = A050146(n+1).

Cf. A337992 (row sums).

Sequence in context: A125133 A109310 A219190 * A120391 A064217 A026623

Adjacent sequences: A337614 A337615 A337616 * A337618 A337619 A337620

Keyword

nonn,tabl

Author

Peter Luschny, Oct 19 2020

Status

approved