A388203 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(n+3,k).

1, 13, 121, 1000, 7834, 59719, 448417, 3337492, 24708190, 182319418, 1342582762, 9874401376, 72571613716, 533160730795, 3916383099889, 28768408020316, 211347150876598, 1552952656383862, 11413589936099950, 83907782844335248, 617030596863871276

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+4).

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

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

G.f.: 1/(sqrt(1-8*x+4*x^2) * ((1-2*x + sqrt(1-8*x+4*x^2))/2)^3).

D-finite with recurrence n*(n+3)*a(n) +2*(-3*n^2-10*n-13)*a(n-1) +4*(-3*n^2+4*n-6)*a(n-2) +8*(n+1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 16 2025

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

Mathematica

Table[Sum[ 3^k*Binomial[ n, k]*Binomial[n+3, 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+3, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(n+3, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 21 2025

Crossrefs

Cf. A069835, A388047, A388202, A388204.

Sequence in context: A297394 A014918 A081033 * A091111 A196921 A317483

Adjacent sequences: A388200 A388201 A388202 * A388204 A388205 A388206

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved