A386882 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n+4,k).

1, 11, 85, 575, 3649, 22363, 134245, 795455, 4673345, 27298155, 158819253, 921406335, 5334940545, 30845044155, 178153920965, 1028224765823, 5931412422529, 34203940374475, 197194672920085, 1136725029892031, 6552204905872321, 37767192179496731, 217698278506411045

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1-x)^n/(1-2*x)^(n+5).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(n+k+4,k).

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k+4,n).

G.f.: 1/(sqrt(1-6*x+x^2) * ((1-x + sqrt(1-6*x+x^2))/2)^4).

a(n) = [x^n] (1+x)^(n+4) * (2+x)^n.

D-finite with recurrence n*(n+4)*a(n) +5*(-n^2-4*n-6)*a(n-1) +5*(-n^2-2)*a(n-2) +(n-2)*(n+2)*a(n-3)=0. - R. J. Mathar, Sep 26 2025

Mathematica

Table[Sum[ 2^k* Binomial[ n, k]*Binomial[n+4, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 23 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n+4, k));
(Magma) [&+[2^k*Binomial(n, k)*Binomial(n+4, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2025

Crossrefs

Column k=4 of A113139.

Cf. A006321, A008288.

Sequence in context: A379588 A271558 A295168 * A001240 A129180 A082365

Adjacent sequences: A386879 A386880 A386881 * A386883 A386884 A386885

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved