A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2*k,2*n-2*k).

1, 8, 44, 336, 2166, 11832, 60576, 292248, 1334817, 5840296, 24637976, 100684376, 400255050, 1553016960, 5897388492, 21967711160, 80425346844, 289868771928, 1029979010972, 3612517052608, 12520285820362, 42919328903928, 145643017892472, 489606988741128

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (16, -104, 352, -708, 1232, -2800, 5168, -6122, 9728, -18008, 16816, -19152, 37744, -27096, 24960, -50611, 24960, -27096, 37744, -19152, 16816, -18008, 9728, -6122, 5168, -2800, 1232, -708, 352, -104, 16, -1).

Formula

G.f.: (Sum_{k=0..4} 4^k * binomial(8,2*k) * (1-x-x^2)^(8-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^8.

Mathematica

Table[Sum[Binomial[k+7, 7]*Binomial[2*k, 2*n-2*k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Apr 22 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(2*k, 2*n-2*k));
(PARI) my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
(Magma) [&+[Binomial(k+7, 7) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025

Crossrefs

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382473.

Cf. A034839, A377159.

Sequence in context: A377159 A210118 A112908 * A264819 A350128 A028565

Adjacent sequences: A382471 A382472 A382473 * A382475 A382476 A382477

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 28 2025

Status

approved