A365193 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^3).

1, 1, 6, 49, 463, 4760, 51702, 583712, 6781774, 80555066, 973813974, 11941861079, 148191437719, 1857464450449, 23481830726334, 299056887494427, 3833349330581255, 49416395972195630, 640256115370243620, 8332835556325119938, 108890550249605779116

Offset

0,3

Links

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

Formula

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

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

Mathematica

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

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365192, A365194.

Cf. A365187.

Sequence in context: A390415 A388133 A371406 * A365188 A143165 A008786

Adjacent sequences: A365190 A365191 A365192 * A365194 A365195 A365196

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2023

Status

approved