A391307 Expansion of g^3/(1 + x^2*g^4), where g = 1+x*g^3 is the g.f. of A001764.

1, 3, 11, 48, 232, 1194, 6411, 35496, 201173, 1161289, 6803972, 40356280, 241842772, 1462062606, 8906041147, 54609234320, 336793683739, 2087812039797, 13001938411011, 81303761475288, 510299694531264, 3213679925601912, 20300791714902036, 128600686980843168

Offset

0,2

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+3) * binomial(3*n-2*k+3,n-2*k)/(3*n-2*k+3).

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

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

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

Mathematica

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

Crossrefs

Cf. A391301, A391306.

Cf. A001764.

Sequence in context: A126180 A121139 A316703 * A362741 A192399 A233162

Adjacent sequences: A391304 A391305 A391306 * A391308 A391309 A391310

Keyword

nonn

Author

Seiichi Manyama, Dec 06 2025

Status

approved