A397513 - OEIS

%I #17 Jun 29 2026 11:31:12

%S 1,5,65,951,14651,232407,3756831,61534891,1017777761,16960532172,

%T 284320206391,4789320531464,80998807949865,1374509101940902,

%U 23391969665137355,399085386227367204,6823496869562499259,116890000259795561005,2005779247640013424776

%N a(n) = Sum_{k=0..n} binomial(5*n+2*k-3,k).

%F G.f.: 1/(g^3 * (1-7*x*g^6) * (1-x*g^5)) where g = 1+x*g^7 is the g.f. of A002296.

%F G.f.: 1/((7-6*g) * (1-g+g^2)) where g = 1+x*g^7 is the g.f. of A002296.

%F Here and below, binomial(N,k) = 0 for k<0.

%F a(n) = Sum_{k=0..n} (-1)^k * binomial(7*n+k-1,n-k).

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(7*n-k-2,n-2*k).

%F This is the special case l=5, m=2, c=0, r=1, s=0 of the following family. For integers l, m, c in Z, and for any constants r and s, define a_{l,m,c,r,s}(n) = Sum_{k=0..n} (r*binomial(l*n+m*k-m-1+c,k) - s*binomial(l*n+m*k-m-1+c,k-1)). A_{l,m,c,r,s}(x) = Sum_{n>=0} a_{l,m,c,r,s}(n)*x^n = t^c*(r+s-s*t)/((l+m-(l+m-1)*t)*(1-t+t^m)), where t = t(x) satisfies t = 1 + x*t^(l+m), equivalently y = x*(1+y)^(l+m) with y=t-1.

%o (PARI) a(n) = sum(k=0, n, binomial(5*n+2*k-3, k));

%Y Cf. A397509, A397510, A397511, A397512.

%Y Cf. A002296, A397516.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Jun 28 2026