A370280 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.

1, 3, 25, 234, 2305, 23373, 241486, 2527920, 26720529, 284555700, 3048323135, 32812937820, 354619072990, 3845377105794, 41817926091120, 455893204069944, 4980851709418353, 54521955043418925, 597823622561048020, 6564929893462467450, 72189820135528858455

Offset: 0

Author

Seiichi Manyama, Feb 13 2024

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..940

Crossrefs

Cf. A069723, A249924, A370282.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 100);
A370280:= func< n | Coefficient(R!( 1/(1-3*x+x^2)^n ), n) >;
[A370280(n): n in [0..30]]; // G. C. Greubel, Feb 07 2025

Mathematica

A370280[n_]:= Coefficient[Series[1/(1-3*x+x^2)^n, {x,0,100}], x, n];
Table[A370280[n], {n,0,40}] (* G. C. Greubel, Feb 07 2025 *)

PARI

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(3*n+k-1, n-k));

SageMath

def A370280(n): return sum(binomial(n+j-1,j)*binomial(3*n+j-1,n-j) for j in range(n+1))
print([A370280(n) for n in range(31)]) # G. C. Greubel, Feb 07 2025

Formula

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

a(n) ~ sqrt((4 + sqrt(6))/(24*Pi*n)) * ((27 + 12*sqrt(6))/5)^n. - Vaclav Kotesovec, Feb 07 2025