A388207 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * binomial(n+4,k).

1, 21, 289, 3333, 35073, 350005, 3379489, 31933221, 297326337, 2739788117, 25057889313, 227913373253, 2064391824897, 18639827016309, 167893750058529, 1509403565177445, 13549789700813313, 121493423266590357, 1088358864961303329, 9742576323085181829, 87161303214793783041

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1-3*x)^n/(1-4*x)^(n+5).

a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n,k) * binomial(n+k+4,k).

a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(n+k+4,n).

G.f.: 1/(sqrt(1-10*x+9*x^2) * ((1-3*x + sqrt(1-10*x+9*x^2))/2)^4).

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

Mathematica

Table[Sum[4^k*Binomial[n, k]*Binomial[n+4, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 17 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 4^k*binomial(n, k)*binomial(n+4, k));
(Magma) [&+[4^k*Binomial(n, k)*Binomial(n+4, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 17 2025

Crossrefs

Cf. A084771, A388060, A388205, A388206.

Sequence in context: A317042 A317696 A025967 * A022452 A183731 A101703

Adjacent sequences: A388204 A388205 A388206 * A388208 A388209 A388210

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved