A391220 Expansion of 1/(g * (2-g))^2, where g = 1+x*g^4 is the g.f. of A002293.

1, 0, 2, 16, 123, 960, 7640, 61872, 508529, 4231936, 35590392, 302015040, 2582784897, 22236918336, 192587900648, 1676695299104, 14665645756365, 128813057698816, 1135667256787064, 10046643668671680, 89153113127499692, 793380999016226560, 7078730315047506528

Offset

0,3

Links

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

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A391209.

G.f.: 1/(1 - x^2*g^8)^2, where g = 1+x*g^4 is the g.f. of A002293.

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

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

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

Mathematica

Join[{1}, Table[Sum[(4/n)*Binomial[k+1, 2]*Binomial[4*n, n-2*k], {k, 0, Floor[n/2]}], {n, 1, 25}]] (* Vincenzo Librandi, Dec 04 2025 *)

Prog

(PARI) a(n) = if(n==0, 1, 4/n*sum(k=0, n\2, binomial(k+1, 2)*binomial(4*n, n-2*k)));
(Magma) [1] cat [(4 / n) * &+[Binomial(k+1, 2) * Binomial(4*n, n-2*k): k in [0..Floor(n/2)]] : n in [1..30] ]; // Vincenzo Librandi, Dec 04 2025

Crossrefs

Cf. A391209, A391221, A391224.

Cf. A002293.

Sequence in context: A229808 A301947 A027309 * A241466 A069868 A338186

Adjacent sequences: A391217 A391218 A391219 * A391221 A391222 A391223

Keyword

nonn

Author

Seiichi Manyama, Dec 04 2025

Status

approved