A365192 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^2).

1, 1, 6, 48, 443, 4445, 47107, 518835, 5880223, 68130860, 803369481, 9609294542, 116310009888, 1421951861817, 17533301767624, 217796367181117, 2722942699583650, 34236790400004432, 432649744252128084, 5492060945760586212, 69998993052214823013

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Formula

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

G.f.: 1 + Series_Reversion( x / ((1+x)^2 * (x+(1+x)^3)) ). - Seiichi Manyama, Sep 28 2025

Mathematica

a[n_]:=Sum[Binomial[2*n+3*k+1, k]*Binomial[n-1, n-k]/(2*n+3*k+1), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Vincenzo Librandi, Oct 22 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(n-1, n-k)/(2*n+3*k+1));
(Magma) [&+[Binomial(2*n+3*k+1, k) * Binomial(n-1, n-k)/(2*n+3*k+1) : k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Oct 22 2025

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365193, A365194.

Cf. A365186.

Sequence in context: A085457 A188911 A364923 * A365187 A378155 A319292

Adjacent sequences: A365189 A365190 A365191 * A365193 A365194 A365195

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2023

Status

approved