A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.
1, 4, 38, 428, 5204, 66104, 863840, 11515308, 155779966, 2131436392, 29426804398, 409254436452, 5726378247412, 80535621269208, 1137609359823936, 16130112288879248, 229462608491483364, 3273749607191060480, 46826932120849617128, 671341041479214814160, 9644654058165119642624
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..850
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] - 1)/3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)
Formula
Sum_{k=0..n} a(k) * a(n-k) = A385497(n).
G.f.: 1/sqrt(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g/sqrt((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
a(n) ~ 2^(6*n - 1/2) * 3^(6*n + 3/4) / (Gamma(1/4) * n^(3/4) * 5^(5*n + 1/4)) * (1 + 7*Gamma(1/4)^2/(48*Pi*sqrt(30*n))). - Vaclav Kotesovec, Aug 20 2025