A397567 - OEIS

%I #10 Jul 01 2026 10:06:49

%S 1,5,54,664,8625,115506,1577311,21833033,305238726,4300158499,

%T 60947481440,868075247727,12414285158328,178143045874721,

%U 2563779890825235,36989976295984049,534858504103606897,7748732574979636374,112451247125555173765,1634406098930351324291

%N a(n) = Sum_{k=0..n} binomial(4*n+2*k-2,k).

%F G.f.: 1/(g^2 * (1-6*x*g^5) * (1-x*g^4)) where g = 1+x*g^6 is the g.f. of A002295.

%F G.f.: g/((6-5*g) * (1-g+g^2)) where g = 1+x*g^6 is the g.f. of A002295.

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

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

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

%F This is the special case l=4, m=2, c=1, 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(4*n+2*k-2, k));

%Y Cf. A397512, A396691, A397568.

%Y Cf. A002295.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Jul 01 2026