A365194 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).

1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

a(n) = Sum_{k=0..n} binomial(6*n-k+1,k) * binomial(n-1,n-k)/(6*n-k+1).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*n+2*k+1,k) * binomial(n-1,n-k)/(5*n+2*k+1).

a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(6*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Dec 26 2024

G.f.: 1 + Series_Reversion( x / ((1+x)^5 * (1+x*(1+x))) ). - Seiichi Manyama, Oct 07 2025

Mathematica

a[n_]:=Sum[(-1)^(n-k)*Binomial[5*n+2*k+1, k]*Binomial[n-1, n-k]/(5*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(6*n-k+1, k)*binomial(n-1, n-k)/(6*n-k+1));
(Magma) [&+[(-1)^(n-k) * Binomial(5*n+2*k+1, k) * Binomial(n-1, n-k)/(5*n+2*k+1) : k in [0..n]] : n in [0..30]]; // Vincenzo Librandi, Oct 22 2025

Crossrefs

Cf. A002295, A243667, A349332, A364748, A365192, A365193.

Cf. A219537, A271469, A364765.

Sequence in context: A357154 A390384 A388538 * A365755 A232821 A355162

Adjacent sequences: A365191 A365192 A365193 * A365195 A365196 A365197

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2023

Status

approved