A397565 - OEIS

%I #10 Jul 01 2026 09:59:52

%S 1,3,34,428,5628,75926,1041876,14471417,202839843,2863267846,

%T 40646324515,579673788822,8298771433240,119193890037793,

%U 1716729717777120,24785397074603618,358595277154362550,5197804355574056304,75466114062838427516,1097299074257879775625

%N a(n) = Sum_{k=0..n} binomial(3*n+3*k-4,k).

%F G.f.: 1/(g^4 * (1-6*x*g^5) * (1-x*g^3)) where g = 1+x*g^6 is the g.f. of A002295.

%F G.f.: 1/((6-5*g) * (1-g+g^3)) where g = 1+x*g^6 is the g.f. of A002295.

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

%F This is the special case l=3, m=3, 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(3*n+3*k-4, k));

%Y Cf. A397512, A397537, A397539.

%Y Cf. A002295.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Jul 01 2026