A372417 Coefficient of x^n in the expansion of ( (1-x+x^3)^2 / (1-x)^3 )^n.

1, 1, 3, 16, 75, 336, 1536, 7155, 33627, 158974, 755508, 3606648, 17281776, 83068766, 400368741, 1934204661, 9363509531, 45411373098, 220593832062, 1073127878085, 5227288727580, 25492636911240, 124457166046832, 608207193661734, 2974913417047440

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^3)^2 ). See A369231.

Prog

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

Crossrefs

Cf. A369231, A372416.

Sequence in context: A335625 A317365 A207836 * A005947 A003769 A005386

Adjacent sequences: A372414 A372415 A372416 * A372418 A372419 A372420

Keyword

nonn

Author

Seiichi Manyama, Apr 29 2024

Status

approved