A387277 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+4,k+4) * binomial(2*k+8,k+8).

1, 25, 381, 4585, 47978, 458010, 4100370, 35027850, 288845370, 2317794050, 18203687502, 140533725150, 1069904389008, 8052575725680, 60033791987424, 444015014417280, 3261950250436845, 23827019766988725, 173193081555808545, 1253583401573658925, 9040278899072328006

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

n*(n+8)*a(n) = (n+4) * (5*(2*n+7)*a(n-1) - 21*(n+3)*a(n-2)) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+4,n-2*k) * binomial(2*k+4,k).

a(n) = [x^n] (1+5*x+x^2)^(n+4).

E.g.f.: exp(5*x) * BesselI(4, 2*x), with offset 4.

Mathematica

Table[Sum[3^(n-k)*Binomial[n+4, k+4]*Binomial[2*k+8, k+8], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 30 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n+4, k+4)*binomial(2*k+8, k+8));
(Magma) [&+[3^(n-k) * Binomial(n+4, k+4) * Binomial(2*k+8, k+8): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025

Crossrefs

Cf. A387275, A387276, A387278.

Cf. A387274.

Sequence in context: A225968 A094190 A069396 * A228216 A261485 A387283

Adjacent sequences: A387274 A387275 A387276 * A387278 A387279 A387280

Keyword

nonn

Author

Seiichi Manyama, Aug 24 2025

Status

approved