A388133 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(2*n+1,k).

1, 6, 49, 440, 4131, 39810, 390300, 3873504, 38794135, 391287386, 3968936049, 40443812744, 413708253076, 4245636082680, 43691381448264, 450704409453504, 4659076810596159, 48251654248057050, 500540355381945795, 5200020390043329480, 54093605141067977499

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..950

Formula

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

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,k).

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(2*n+k+1,n).

D-finite with recurrence 18*n*(2*n+1)*a(n) +3*(-159*n^2+70*n-19)*a(n-1) +(1013*n^2-2799*n+1996)*a(n-2) -112*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 16 2025

a(n) = [x^n] (1+x)^(2*n+1) * (1+3*x)^n. - Seiichi Manyama, Sep 21 2025

Recurrence (of order 2): 6*n*(2*n + 1)*(25*n - 16)*a(n) = (3275*n^3 - 2096*n^2 - 477*n + 270)*a(n-1) - 16*(n-1)*(2*n - 1)*(25*n + 9)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Mathematica

Table[Sum[ 3^(n-k)* Binomial[ n, k]*Binomial[2*n+1, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 24 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n, k)*binomial(2*n+1, k));
(Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(2*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 24 2025

Crossrefs

Cf. A098663, A388134, A388135.

Cf. A387929, A388048.

Sequence in context: A098306 A055847 A390415 * A371406 A365193 A365188

Adjacent sequences: A388130 A388131 A388132 * A388134 A388135 A388136

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved