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

Author

Peter Luschny, Nov 11 2022

Keywords

• nonn

Crossrefs

Cf. A000108, A358119.

Programs

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);

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

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