A349311 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^5) / (1 - x).

1, 2, 12, 112, 1232, 14832, 189184, 2512064, 34358784, 480745984, 6848734464, 99003237376, 1448575666176, 21411827808256, 319255531155456, 4796005997940736, 72520546008219648, 1102912584949792768, 16859182461720526848, 258886644574700699648

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(5*k,k) / (4*k+1).

a(n) = F([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [1/2, 3/4, 1, 5/4], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 + 4*r) / (2 * 5^(3/4) * sqrt(2*Pi) * (1-r)^(1/4) * n^(3/2) * r^(n + 1/4)), where r = 0.0600920016324256496641829206872407657377702010870270617... is the real root of the equation 4^4 * (1-r)^5 = 5^5 * r. - Vaclav Kotesovec, Nov 15 2021

Mathematica

nmax = 19; A[_] = 0; Do[A[x_] = (1 + x A[x]^5)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A002294, A006318, A346626, A346647, A349310, A349312, A349313, A349314.

Sequence in context: A047855 A199045 A009232 * A218222 A355112 A374562

Adjacent sequences: A349308 A349309 A349310 * A349312 A349313 A349314

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 14 2021

Status

approved