A391678 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(k+2,2) * binomial(k,n-k)^2.

1, 6, 42, 368, 2616, 18672, 132112, 906624, 6151536, 41243552, 273427680, 1797219840, 11722297600, 75943488000, 489123171840, 3133819805696, 19984972283136, 126915902664192, 802948305023488, 5062568720879616, 31819873105102848, 199427697561079808

Offset

0,2

Links

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

Formula

G.f.: ((1-2*x-6*x^2)^2 + 24*x^3)/((1-2*x-6*x^2)^2 - 48*x^3)^(5/2).

Mathematica

Table[Sum[2^k*3^(n-k)*Binomial[k+2, 2]*Binomial[k, n-k]^2, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(k+2, 2)*binomial(k, n-k)^2);
(Magma) [&+[2^k*3^(n-k)*Binomial(k+2, 2)*Binomial(k, n-k)^2: k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 19 2025

Crossrefs

Cf. A108490, A391677.

Sequence in context: A387083 A052589 A074107 * A394440 A187121 A225497

Adjacent sequences: A391675 A391676 A391677 * A391679 A391680 A391681

Keyword

nonn

Author

Seiichi Manyama, Dec 16 2025

Status

approved