A391223 Expansion of 1/(g * (2-g))^3, where g = 1+x*g^2 is the g.f. of A000108.

1, 0, 3, 12, 48, 192, 769, 3084, 12381, 49744, 199977, 804276, 3235668, 13020192, 52400538, 210909656, 848950359, 3417275616, 13755595689, 55369756356, 222871001064, 897051906816, 3610447601694, 14530522785576, 58475788263354, 235311984533472, 946856171141034

Offset

0,3

Links

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

Formula

G.f.: 1/(1 - x^2*g^4)^3, where g = 1+x*g^2 is the g.f. of A000108.

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

a(n) = (6/n) * Sum_{k=0..n-1} binomial(k+2,3) * binomial(2*n-4,n-1-k) for n > 0.

a(n) = (6/n) * Sum_{k=0..floor(n/2)} binomial(k+2,3) * binomial(2*n,n-2*k) for n > 0.

a(n) = 3*4^(n-2)/2 + binomial(2*n-2,n) + (n-1)*binomial(2*n-4,n-2)/2 for n > 1.

E.g.f.: (39 + 9*exp(4*x) - 36*x + 8*exp(2*x)*(2*(3 + (x - 6)*x)*BesselI(0, 2*x) + (13 - 2*x)*x*BesselI(1, 2*x)))/96. - Stefano Spezia, Dec 04 2025

Mathematica

Join[{1}, Table[Sum[(6/n)*Binomial[k+2, 3]*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, 3*4^(n-2)/2+binomial(2*n-2, n)+(n-1)*binomial(2*n-4, n-2)/2);
(Magma) [1] cat [(6 / n) * &+[Binomial(k+2, 3) * Binomial(2*n, n-2*k): k in [0..Floor(n/2)]] : n in [1..30] ]; // Vincenzo Librandi, Dec 04 2025

Crossrefs

Cf. A001791, A391222.

Cf. A000108.

Sequence in context: A077828 A002001 A164346 * A113956 A323261 A103943

Adjacent sequences: A391220 A391221 A391222 * A391224 A391225 A391226

Keyword

nonn

Author

Seiichi Manyama, Dec 04 2025

Status

approved