A397514 a(n) = Sum_{k=0..n} binomial(3*n+2*k-3,k-1).

0, 1, 8, 77, 799, 8625, 95381, 1072072, 12191464, 139863218, 1615571749, 18764341805, 218924287555, 2563800729881, 30119987939735, 354820919500737, 4189771263878559, 49575855774030751, 587683571695829594, 6977862787102136461, 82972236113530966223, 987897037991665663841

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f.: x*g^2/((1-5*x*g^4) * (1-x*g^3)) where g = 1+x*g^5 is the g.f. of A002294.

G.f.: (-1+g)/((5-4*g) * (1-g+g^2)) where g = 1+x*g^5 is the g.f. of A002294.

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(5*n+k-1,n-k-1).

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(5*n-k-2,n-2*k-1).

This is the special case l=3, m=2, c=0, r=0, s=-1 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.

Prog

(PARI) C(n, k) = if(k>=0, binomial(n, k));
a(n) = sum(k=0, n, (-1)^k*C(5*n+k-1, n-k-1));

Crossrefs

Cf. A397515, A397516.

Cf. A002294, A397511.

Sequence in context: A091686 A091691 A024103 * A187638 A092661 A196071

Adjacent sequences: A397511 A397512 A397513 * A397515 A397516 A397518

Keyword

nonn

Author

Seiichi Manyama, Jun 28 2026

Status

approved