A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

1, 4, 2, 16, 16, 5, 64, 96, 60, 14, 256, 512, 480, 224, 42, 1024, 2560, 3200, 2240, 840, 132, 4096, 12288, 19200, 17920, 10080, 3168, 429, 16384, 57344, 107520, 125440, 94080, 44352, 12012, 1430, 65536, 262144, 573440, 802816, 752640, 473088, 192192, 45760, 4862

Offset: 0

Author

Peter Luschny, Feb 03 2015

Keywords

• nonn
• tabl

Examples

			[   1]
[   4,     2]
[  16,    16,     5]
[  64,    96,    60,    14]
[ 256,   512,   480,   224,    42]
[1024,  2560,  3200,  2240,   840,  132]
[4096, 12288, 19200, 17920, 10080, 3168, 429]
		

Crossrefs

Cf. A108198 (Peter Bala), A000302, A000108, A025230, A002699, A128650, A254633.

Programs

Maple

h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);

Mathematica

T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019 *)

Sage

A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)
for n in range(7): [A254632(n,k) for k in (0..n)]

Formula

T(n,0) = A000302(n).

T(n,n) = A000108(n+1).

T(n,1) = A002699(n) for n>=1.

T(n,n-1) = A128650(n+2) for n>=1.

T(2*n,n) = A254633(n).

T(n,k) = 4^(n-k)*C(n,k)*Catalan(k+1).

sum(k=0..n, T(n,k)) = A025230(n+2).