A391222 Expansion of 1/(g * (2-g))^2, where g = 1+x*g^2 is the g.f. of A000108.

1, 0, 2, 8, 31, 120, 466, 1816, 7099, 27824, 109294, 430104, 1695222, 6690448, 26434916, 104551024, 413858131, 1639464544, 6498928726, 25777365784, 102297477346, 406160315344, 1613303235196, 6410662911184, 25482557158366, 101326443713440, 403023636746956

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/(1 - x^2*g^4)^2, where g = 1+x*g^2 is the g.f. of A000108.

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

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

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

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

a(n) = 4^(n-2) + binomial(2*n-2,n) for n > 1.

E.g.f.: (7 + exp(4*x) - 4*x + 8*exp(2*x)*((1 - 2*x)*BesselI(0, 2*x) + 2*x*Bessel(1, 2*x)))/16. - Stefano Spezia, Dec 04 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A391219, A391224.

Cf. A001791, A391223.

Cf. A000108.

Sequence in context: A389157 A216318 A018916 * A281831 A206229 A027073

Adjacent sequences: A391219 A391220 A391221 * A391223 A391224 A391225

Keyword

nonn

Author

Seiichi Manyama, Dec 04 2025

Status

approved