A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).

1, 2, 5, 22, 68, 206, 631, 1870, 5467, 15836, 45416, 129260, 365565, 1028122, 2877697, 8021010, 22274476, 61653850, 170152275, 468347046, 1286055927, 3523777912, 9635982160, 26302324504, 71674754873, 195015074610, 529846108989, 1437657038030, 3896050721940

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-11,0,-2,4,-1).

Formula

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

a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).

Mathematica

Table[Sum[(k+1)*Binomial[2*k, 2*n-2*k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Apr 23 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*binomial(2*k, 2*n-2*k));
(PARI) my(N=1, 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) [&+[(k+1) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Crossrefs

Cf. A108479, A382230, A382470, A382471, A382472, A382473, A382474.

Cf. A034839.

Sequence in context: A041165 A041006 A346557 * A288028 A083465 A339646

Adjacent sequences: A381418 A381419 A381420 * A381422 A381423 A381424

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 28 2025

Status

approved