A383757 Expansion of 1/Product_{k=0..3} (1 - 3^k * 4^(3-k) * x).

1, 175, 19525, 1776775, 144142141, 10884484975, 783802527925, 54630820881175, 3721247723926381, 249337226367003775, 16508103305566548325, 1083453420457687217575, 70652392978007927384221, 4585369275138131990546575, 296541443098920894741800725, 19127262646595562017053105975

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Index entries for linear recurrences with constant coefficients, signature (175,-11100,302400,-2985984).

Formula

a(n) = A383755(n+3,3).

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

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

a(n) = (-27^(n+2) + 111*36^(n+1) - 148*48^(n+1) + 64^(n+2))/259.

a(n) = 175*a(n-1) - 11100*a(n-2) + 302400*a(n-3) - 2985984*a(n-4).

Prog

(PARI) a(n) = (-27^(n+2)+111*36^(n+1)-148*48^(n+1)+64^(n+2))/259;
(SageMath)
def a(n): return 3^(3*n)*q_binomial(n+3, 3, 4/3)

Crossrefs

Cf. A383755.

Sequence in context: A239429 A268872 A104651 * A212946 A172685 A344280

Adjacent sequences: A383754 A383755 A383756 * A383758 A383759 A383760

Keyword

nonn,easy

Author

Seiichi Manyama, May 09 2025

Status

approved