A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2*k,2*n-2*k).

1, 4, 14, 80, 345, 1336, 5074, 18404, 64460, 220276, 736242, 2415128, 7798043, 24833160, 78131242, 243211412, 749926963, 2292771088, 6956262660, 20959406680, 62753991192, 186809711448, 553172044548, 1630068765840, 4781871397429, 13969460520764

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-26,88,-48,24,-163,24,-48,88,-26,16,-20,8,-1).

Formula

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

a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) - 26*a(n-4) + 88*a(n-5) - 48*a(n-6) + 24*a(n-7) - 163*a(n-8) + 24*a(n-9) - 48*a(n-10) + 88*a(n-11) - 26*a(n-12) + 16*a(n-13) - 20*a(n-14) + 8*a(n-15) - a(n-16).

Mathematica

Table[Sum[Binomial[k+3, 3]*Binomial[2*k, 2*n-2*k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Apr 10 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(2*k, 2*n-2*k));
(PARI) my(N=3, 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+3, 3) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025

Crossrefs

Cf. A108479, A381421, A382230, A382471, A382472, A382473, A382474.

Cf. A034839, A377148.

Sequence in context: A186638 A187847 A277039 * A003707 A355950 A063862

Adjacent sequences: A382467 A382468 A382469 * A382471 A382472 A382473

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 28 2025

Status

approved