A390170 G.f. A(x) satisfies A(x) = 1 + x/(1+x^2)^2 * A(x)^3.

1, 1, 3, 10, 43, 204, 1018, 5270, 28047, 152495, 843396, 4729896, 26834766, 153740975, 888209382, 5168763962, 30269667603, 178259462436, 1054995326733, 6271530654926, 37430712075972, 224206609242404, 1347384314733296, 8121439960283682, 49086762410196710, 297433555174509982

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 * binomial(2*n-3*k-1,k) * A001764(n-2*k).

Mathematica

Table[Sum[(-1)^k*Binomial[2*n-3*k-1, k]*Binomial[3*(n-2*k), n-2*k]/(2*(n-2*k)+1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 14 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-3*k-1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));
(Magma) [&+[(-1)^k*Binomial(2*n-3*k-1, k)*Binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 14 2025

Crossrefs

Cf. A367059, A385065, A385317, A390169.

Cf. A001764, A389284.

Sequence in context: A151085 A082936 A389942 * A347006 A205487 A323667

Adjacent sequences: A390167 A390168 A390169 * A390171 A390172 A390173

Keyword

nonn

Author

Seiichi Manyama, Oct 28 2025

Status

approved