A389292 a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(2*k,n-2*k).

1, 0, 2, 6, 14, 60, 182, 616, 2166, 7260, 25322, 87802, 304850, 1067248, 3734152, 13109856, 46130214, 162540060, 573823712, 2028534772, 7180402614, 25447716580, 90282520770, 320619948480, 1139645397930, 4054224741060, 14433781060584, 51423341981928

Offset

0,3

Links

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

Formula

a(n) = [x^n] 1/(1 - x^2 * (1 + x)^2)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1 - x^2 * (1 + x)^2) ). See A389294.

Mathematica

Table[Sum[Binomial[n+k-1, k]Binomial[2*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n+k-1, k)*binomial(2*k, n-2*k));
(Magma) [&+[Binomial(n+k-1, k) * Binomial(2*k, n-2*k) : k in [0..Floor(n/2)] ]: n in [0..30]]; // Vincenzo Librandi, Oct 06 2025

Crossrefs

Cf. A362087, A389294.

Sequence in context: A295974 A324365 A346208 * A354533 A192764 A055691

Adjacent sequences: A389289 A389290 A389291 * A389293 A389294 A389295

Keyword

nonn

Author

Seiichi Manyama, Sep 28 2025

Status

approved