A389128 a(n) = Sum_{k=0..floor(n/4)} binomial(n,k) * binomial(n+k,n-4*k).

1, 1, 1, 1, 5, 31, 127, 393, 1037, 2665, 7591, 24443, 81731, 264837, 820275, 2478841, 7522349, 23301765, 73463257, 232730545, 733726155, 2298299319, 7181056499, 22484743007, 70688617451, 222916145131, 703639618241, 2220091646761, 7001046148591, 22081549833985

Offset

0,5

Comments

Binomial transform of A389126.

Links

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

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389126(k).

a(n) = [x^n] (1 + x + x^4 * (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 + x^4 * (1 + x)^2) ). See A389132.

Mathematica

Table[Sum[Binomial[n, k]*Binomial[n+k, n-4*k], {k, 0, Floor[n/4]}], {n, 0, 30}] (* Vincenzo Librandi, Sep 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(n, k)*binomial(n+k, n-4*k));
(Magma) [&+[Binomial(n, k) * Binomial(n+k, n-4*k): k in [0..Floor(n/4)]]: n in [0..30]]; // Vincenzo Librandi, Sep 26 2025

Crossrefs

Cf. A387125, A389126, A389132.

Sequence in context: A053699 A152122 A260045 * A267938 A041303 A184446

Adjacent sequences: A389125 A389126 A389127 * A389129 A389130 A389131

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2025

Status

approved