A388129 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(2*n+1,k).

1, 5, 34, 253, 1966, 15674, 127100, 1043165, 8639350, 72053110, 604304636, 5091508690, 43061767516, 365373061772, 3108708313744, 26513185631965, 226595976712870, 1940187521222222, 16639766719758764, 142918245750677158, 1229139650429018852, 10583595038040435020

Offset

0,2

Links

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

Formula

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

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

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

D-finite with recurrence 16*n*(2*n+1)*a(n) +4*(-81*n^2+25*n-4)*a(n-1) +(371*n^2-1073*n+782)*a(n-2) -10*(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+2*x)^n. - Seiichi Manyama, Sep 21 2025

Recurrence (of order 2): 4*n*(2*n + 1)*(17*n - 11)*a(n) = (1207*n^3 - 781*n^2 - 162*n + 96)*a(n-1) - 2*(n-1)*(2*n - 1)*(17*n + 6)*a(n-2). - Vaclav Kotesovec, Nov 09 2025

Mathematica

Table[Sum[ 2^(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, 2^(n-k)*binomial(n, k)*binomial(2*n+1, k));
(Magma) [&+[2^(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. A388130, A388131.

Sequence in context: A183415 A066559 A073279 * A080503 A078284 A083987

Adjacent sequences: A388126 A388127 A388128 * A388130 A388131 A388132

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved