A387929 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(2*n,k).

1, 5, 39, 344, 3199, 30645, 299220, 2960760, 29585679, 297877439, 3017111599, 30708223980, 313808941636, 3217710258716, 33089030081496, 341118479196144, 3524293342682415, 36481349157555411, 378274894452958965, 3928283658932364480, 40849790883027537159

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Formula

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

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

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

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

a(n) ~ 2^(5*n+1) / (sqrt(5*Pi*n) * 3^n). - Vaclav Kotesovec, Sep 21 2025

D-finite with recurrence 42*n*(2*n-1)*a(n) +(-1013*n^2+1253*n-450)*a(n-1) +8*(159*n^2-566*n+515)*a(n-2) -128*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 26 2025

6*n*(2*n-1)*(25*n-34)*a(n) = (3275*n^3-7729*n^2+5156*n-972)*a(n-1) - 16*(n-1)*(2*n-3)*(25*n-9)*a(n-2) for n > 1. - Seiichi Manyama, Oct 10 2025

Mathematica

Table[Sum[3^(n-k)*Binomial[n, k]*Binomial[2*n, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 20 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n, k)*binomial(2*n, k));
(Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 20 2025

Crossrefs

Cf. A387931, A387933.

Sequence in context: A390408 A395370 A244039 * A328554 A213233 A115187

Adjacent sequences: A387926 A387927 A387928 * A387930 A387931 A387932

Keyword

nonn

Author

Seiichi Manyama, Sep 13 2025

Status

approved