A372506 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-2*x) )^n.

1, 3, 23, 198, 1795, 16758, 159446, 1537308, 14967843, 146833830, 1449054178, 14369723316, 143072565454, 1429331585724, 14320668653580, 143838879376248, 1447883909314851, 14602334949928710, 147518977428892010, 1492559101878005700, 15121898521185194970

Offset: 0

Author

Seiichi Manyama, May 04 2024

Keywords

• nonn

Crossrefs

Cf. A001700, A069723, A085614, A255688, A372233.

Programs

Maple

A372506 := proc(n)
    add(binomial(n+k-1,k)*binomial(3*n-1,n-k),k=0..n) ;
end proc:
seq(A372506(n),n=0..80) ; # R. J. Mathar, Oct 24 2024

Mathematica

Table[SeriesCoefficient[1/((1 - x)*(1 - 2*x))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2024 *)
Table[Binomial[3*n - 1, n] * Hypergeometric2F1[-n, n, 2*n, -1], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2024 *)

PARI

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

Formula

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

a(n) ~ (1 + sqrt(3)) * 2^(n - 3/2) * 3^((3*n-1)/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 04 2024

D-finite with recurrence 5*n*(n-1)*a(n) +18*(n-1)*(n-3)*a(n-1) +12*(-45*n^2+90*n-22)*a(n-2) -216*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Oct 24 2024