A391306 Expansion of g/(1 + x^2*g^4), where g = 1+x*g^3 is the g.f. of A001764.

1, 1, 2, 7, 31, 152, 790, 4271, 23767, 135221, 782968, 4598804, 27332956, 164081764, 993424142, 6059089903, 37193922431, 229612233883, 1424621638274, 8878771964287, 55559440930991, 348935946074368, 2198725395056744, 13896564568094212, 88073401284955556

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (4*k+1) * binomial(3*n-2*k+1,n-2*k)/(3*n-2*k+1).

a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * (4*k+1) * binomial(3*n-2*k,n-2*k).

a(n) = Sum_{k=0..n} (-1)^(floor((k+1)/2)) * (2*k+2) * binomial(3*n-k+2,n-k)/(3*n-k+2).

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^(floor((k+1)/2)) * (k+1) * binomial(3*n-k+1,n-k).

Mathematica

Table[ Sum[(-1)^k*(4*k+1)*Binomial[3*n-2*k+1, n-2*k]/(3*n-2*k+1), {k, 0, Floor[n/2]}], {n, 0, 21}] (* Vincenzo Librandi, Dec 07 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*(4*k+1)*binomial(3*n-2*k+1, n-2*k)/(3*n-2*k+1));
(Magma) [&+[(-1)^k*(4*k+1)*Binomial(3*n-2*k+1, n-2*k)/(3*n-2*k+1): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 07 2025

Crossrefs

Cf. A387675, A387677, A389697.

Cf. A001764, A391301.

Sequence in context: A391130 A126033 A369622 * A323632 A369265 A369297

Adjacent sequences: A391303 A391304 A391305 * A391307 A391308 A391309

Keyword

nonn

Author

Seiichi Manyama, Dec 06 2025

Status

approved