A388204 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(n+4,k).

1, 16, 172, 1576, 13327, 107704, 846712, 6538984, 49898962, 377631424, 2841043912, 21282493456, 158924064187, 1183932211336, 8804131172992, 65381568835816, 485035381715182, 3595400186709664, 26635363825706632, 197229161079953584, 1459938029383686646

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1-2*x)^n/(1-3*x)^(n+5).

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

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

G.f.: 1/(sqrt(1-8*x+4*x^2) * ((1-2*x + sqrt(1-8*x+4*x^2))/2)^4).

D-finite with recurrence n*(n+4)*a(n) +2*(-3*n^2-13*n-24)*a(n-1) +4*(-3*n^2+n-10)*a(n-2) +8*(n-2)*(n+2)*a(n-3)=0. - R. J. Mathar, Sep 16 2025

a(n) = [x^n] (1+x)^(n+4) * (3+x)^n. - Seiichi Manyama, Sep 21 2025

Mathematica

Table[Sum[ 3^k*Binomial[ n, k]*Binomial[n+4, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 21 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(n+4, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(n+4, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 21 2025

Crossrefs

Cf. A069835, A388047, A388202, A388203.

Sequence in context: A200673 A230510 A238725 * A221789 A018209 A021174

Adjacent sequences: A388201 A388202 A388203 * A388205 A388206 A388207

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved