A383756 Expansion of 1/Product_{k=0..2} (1 - 3^k * 4^(2-k) * x).

1, 37, 925, 19525, 375661, 6828757, 119609725, 2042733925, 34274529421, 567869330677, 9323118394525, 152047784616325, 2467581667044781, 39901653896747797, 643493505828795325, 10356906506162786725, 166444482073618177741, 2671936126059753592117

Offset: 0

Author

Seiichi Manyama, May 09 2025

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (37,-444,1728).

Crossrefs

Cf. A383755.

Programs

PARI

a(n) = (3*9^(n+1)-7*12^(n+1)+4*16^(n+1))/7;

Sage

def a(n): return 3^(2*n)*q_binomial(n+2, 2, 4/3)

Formula

a(n) = A383755(n+2,2).

a(n) = 3^(2*n) * q-binomial(n+2, 2, 4/3).

G.f.: exp( Sum_{k>=1} f(3*k)/f(k) * x^k/k ), where f(k) = 4^k - 3^k.

a(n) = (3*9^(n+1) - 7*12^(n+1) + 4*16^(n+1))/7.

a(n) = 37*a(n-1) - 444*a(n-2) + 1728*a(n-3).