A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

1, 1, 5, 31, 219, 1678, 13570, 114014, 985542, 8708099, 78298727, 714105907, 6590200215, 61427125994, 577456943614, 5468604044500, 52122539760992, 499613409224137, 4813105582181533, 46576519080852235, 452545041339982871, 4413071971740021275, 43177663974461532959

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..985

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(4*k,k) / (3*k+1).

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

Maple

a:= n-> coeff(series(RootOf(1+x*A^4/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2021

Mathematica

nmax = 22; A[_] = 0; Do[A[x_] = 1 + x A[x]^4/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]

Crossrefs

Cf. A002212, A002293, A307678, A317133, A346646 (partial sums), A349332, A349333, A349334, A349335.

Sequence in context: A097146 A143020 A367238 * A059035 A199877 A058309

Adjacent sequences: A349328 A349329 A349330 * A349332 A349333 A349334

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 15 2021

Status

approved