A358119 - OEIS

%I #14 Nov 12 2022 03:02:13

%S 2,3,15,98,750,6359,57939,556896,5578764,57759397,614328561,

%T 6682078770,74071710414,834535805445,9535609593441,110306008352832,

%U 1289937458160684,15231176767392691,181406519662622559,2177471166182909994,26321521760571055830,320222147815305416123

%N 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.

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

%F 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

%p c := n -> binomial(2*n, n)/(n + 1):

%p A358119 := n -> add((-1)^j*binomial(2*n-j, j)*c(n-j)*c(n-j+2), j = 0..n):

%p seq(A358119(n), n = 0 .. 21);

%o (Python)

%o from math import comb

%o 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

%Y Cf. A000108, A358118.

%K nonn

%O 0,1

%A _Peter Luschny_, Nov 11 2022