A372414 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x)^2 )^n.

1, 1, 3, 13, 55, 231, 981, 4215, 18271, 79735, 349843, 1541783, 6820045, 30263689, 134658681, 600578373, 2684116863, 12017803439, 53895617379, 242054324055, 1088530440315, 4900978877115, 22089865194543, 99662269990363, 450049706481181, 2033999993960581

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(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)^2 / (1-x+x^3) ).

Prog

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

Crossrefs

Cf. A372413, A372415.

Cf. A049128.

Sequence in context: A037583 A302757 A093834 * A296045 A370624 A286191

Adjacent sequences: A372411 A372412 A372413 * A372415 A372416 A372417

Keyword

nonn

Author

Seiichi Manyama, Apr 29 2024

Status

approved