A358118 a(n) = Sum_{j=0..n} (-1)^j*binomial(2*n - j, j)*c(n - j + 1)^2, where c(n) is the n-th Catalan number.

1, 3, 14, 94, 728, 6220, 56960, 549412, 5517746, 57235402, 609632284, 6638455152, 73653980850, 830429715762, 9494320306104, 109882451968812, 1285515149382226, 15184272319785322, 180901975355930990, 2171974459996885630, 26260944994924397048, 319547523490735463404

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: ((1-14*x+x^2)^(1/2)*(x+1)^2*hypergeom([-1/2, 3/2],[1],-16*x/(1-14*x+x^2))-x^3-7*x^2-7*x-1)/(4*x^2). - Mark van Hoeij, Nov 11 2022

Maple

c := n -> binomial(2*n, n)/(n + 1):
A358118 := n -> add((-1)^j*binomial(2*n-j, j)*c(n-j+1)^2, j = 0..n):
seq(A358118(n), n = 0..21);

Prog

(Python)
from math import comb
def A358118(n): return sum((-1 if j&1 else 1)*comb((n<<1)-j, j)*(comb(n-j+1<<1, n-j+1)//(n-j+2))**2 for j in range(n+1)) # Chai Wah Wu, Nov 11 2022

Crossrefs

Cf. A000108, A358119.

Sequence in context: A195134 A089462 A088342 * A364629 A074531 A091906

Adjacent sequences: A358115 A358116 A358117 * A358119 A358120 A358121

Keyword

nonn

Author

Peter Luschny, Nov 11 2022

Status

approved