A358119 a(n) = Sum_{j=0..n} (-1)^j*binomial(2*n - j, j)*c(n - j)*c(n - j + 2), where c(n) is the n-th Catalan number.

2, 3, 15, 98, 750, 6359, 57939, 556896, 5578764, 57759397, 614328561, 6682078770, 74071710414, 834535805445, 9535609593441, 110306008352832, 1289937458160684, 15231176767392691, 181406519662622559, 2177471166182909994, 26321521760571055830, 320222147815305416123

Offset: 0

Author

Peter Luschny, Nov 11 2022

Keywords

• nonn

Crossrefs

Cf. A000108, A358118.

Programs

Maple

c := n -> binomial(2*n, n)/(n + 1):
A358119 := n -> add((-1)^j*binomial(2*n-j, j)*c(n-j)*c(n-j+2), j = 0..n):
seq(A358119(n), n = 0 .. 21);

Python

from math import comb
def A358119(n): return sum((-1 if j&1 else 1)*comb((n<<1)-j,j)*comb(n-j<<1,n-j)*comb(n-j+2<<1,n-j+2)//(n-j+1)//(n-j+3) for j in range(n+1)) # Chai Wah Wu, Nov 11 2022

Formula

a(n) = 2*(-1)^n*hypergeom([5/2, -n, n + 1], [2, 4], 4).

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