A370104 - OEIS

%I #13 Feb 13 2024 10:56:24

%S 1,3,25,219,2025,19253,186469,1829565,18124521,180886260,1815946275,

%T 18318160358,185518492965,1885157971596,19211066004995,

%U 196258973605094,2009302383218409,20610411795602760,211768072490024440,2179156980022097775,22454554231950998275

%N a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(6*n-k-1,n-k).

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

%F The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^5 ). See A365856.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(4*n-2*k-1,n-2*k).

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k-1, k)*binomial(6*n-k-1, n-k));

%o (PARI) a(n, s=2, t=2, u=3) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));

%Y Cf. A351857, A370103.

%Y Cf. A365856.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 10 2024