A372233 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^2) )^n.

1, 2, 12, 77, 520, 3612, 25557, 183192, 1325808, 9666635, 70897112, 522472392, 3865669717, 28697325048, 213649228560, 1594540806612, 11926354293792, 89372808145692, 670865679851667, 5043360211505000, 37965778448487120, 286151354441445570, 2159143860124095120

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

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

D-finite with recurrence +575*n*(n-1)*(n-2)*a(n) +40*(n-1)*(n-2)*(125*n-178)*a(n-1) -16*(n-2)*(3272*n^2-5536*n+75)*a(n-2) +8*(-22112*n^3+169392*n^2-450082*n+415827)*a(n-3) +1344*(96*n^3-1328*n^2+5794*n-8139)*a(n-4) +3072*(4*n-15)*(2*n-9)*(4*n-17)*a(n-5)=0. - R. J. Mathar, May 02 2024

a(n) ~ sqrt((1/8 + cos(arccos(sqrt(37)/8)/3)/sqrt(37))/(Pi*n)) / (-2/3 + sqrt(35/18)*cos(arccos(-4537/(560*sqrt(70)))/3))^n. - Vaclav Kotesovec, May 04 2024

Maple

A372233 := proc(n)
    add(binomial(n+k-1, k) * binomial(3*n-k-1, n-2*k), k=0..floor(n/2));
end proc:
seq(A372233(n), n=0..50) ; # R. J. Mathar, May 02 2024

Mathematica

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

Prog

(PARI) a(n, s=2, 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));

Crossrefs

Cf. A213684, A236339, A372458, A372460.

Sequence in context: A121680 A277478 A372410 * A081014 A223771 A370442

Adjacent sequences: A372230 A372231 A372232 * A372234 A372235 A372236

Keyword

nonn,new

Author

Seiichi Manyama, May 02 2024

Status

approved