A387928 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(2*n,k).

1, 7, 79, 1000, 13327, 182887, 2557684, 36245608, 518690095, 7478672005, 108474941239, 1581019989028, 23135973585796, 339707845836340, 5002372764627352, 73846245417530320, 1092507943002790255, 16193897170945413505, 240444441141909184045, 3575491906219771528672

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Formula

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

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

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

a(n) = hypergeom([-2*n, -n], [1], 3). - Stefano Spezia, Sep 13 2025

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

a(n) ~ sqrt(3 + 19/sqrt(33)) * (59 + 11*sqrt(33))^n / (sqrt(Pi*n) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Sep 21 2025

D-finite with recurrence 10*n*(2*n-1)*a(n) +(-327*n^2+407*n-150)*a(n-1) +8*(39*n^2-206*n+227)*a(n-2) +128*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 26 2025

2*n*(2*n-1)*(11*n-14)*a(n) = (649*n^3-1475*n^2+964*n-180)*a(n-1) + 16*(n-1)*(2*n-3)*(11*n-3)*a(n-2) for n > 1. - Seiichi Manyama, Oct 10 2025

Mathematica

Table[Sum[ 3^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^k*binomial(n, k)*binomial(2*n, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 20 2025

Crossrefs

Cf. A069835, A387930, A387932.

Sequence in context: A198973 A127859 A372969 * A014293 A176792 A186377

Adjacent sequences: A387925 A387926 A387927 * A387929 A387930 A387931

Keyword

nonn

Author

Seiichi Manyama, Sep 13 2025

Status

approved