A382613 Expansion of 1/(1 - x*(1 + x)^3)^2.

1, 2, 9, 28, 88, 270, 808, 2386, 6960, 20104, 57607, 163950, 463907, 1306104, 3661248, 10223820, 28452400, 78941412, 218426608, 602886704, 1660329597, 4563175466, 12517834605, 34280427828, 93729509848, 255900484218, 697712467704, 1899912606358, 5167488465184

Offset: 0

Author

Seiichi Manyama, Mar 31 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (2,5,0,-13,-20,-15,-6,-1).

Crossrefs

Cf. A099234, A382614.

Cf. A001629, A362126.

Programs

Magma

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 08 2025

Mathematica

Table[Sum[(k+1)*Binomial[3*k,n-k],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, Apr 08 2025 *)

PARI

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

Formula

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

a(n) = 2*a(n-1) + 5*a(n-2) - 13*a(n-4) - 20*a(n-5) - 15*a(n-6) - 6*a(n-7) - a(n-8).

G.f.: 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2. - Vincenzo Librandi, Apr 08 2025