A382543 Expansion of 1/(1 - x/(1 - 9*x)^(1/3))^2.

1, 2, 9, 58, 428, 3360, 27295, 226538, 1907889, 16239034, 139326959, 1202856930, 10436521180, 90920984306, 794767853334, 6967126281976, 61224158085137, 539141091531558, 4756357637006941, 42028309478725094, 371898032568193530, 3294977494088601508

Offset: 0

Author

Seiichi Manyama, Mar 31 2025

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A362206, A382544.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 9*x)^(1/3))^2)); // Vincenzo Librandi, May 12 2025

Mathematica

Table[Sum[9^(n-k)* (k+1)* Binomial[n-2*k/3-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 12 2025 *)

PARI

a(n) = sum(k=0, n, 9^(n-k)*(k+1)*binomial(n-2*k/3-1, n-k));

Formula

a(n) = Sum_{k=0..n} 9^(n-k) * (k+1) * binomial(n-2*k/3-1,n-k).

D-finite with recurrence (n-1)*(n-2)*a(n) -3*(n-2)*(11*n-29)*a(n-1) +135*(n-3)*(3*n-10)*a(n-2) +(-2188*n^2+17739*n-35909)*a(n-3) +3*(1466*n^2-14601*n+36292)*a(n-4) +27*(-7*n^2+37*n-42)*a(n-5) +54*(3*n-10)*(3*n-14)*a(n-6)=0. - R. J. Mathar, Apr 02 2025