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A301833
G.f. A(x) satisfies: A(x) = 1/(1 - 2*x*A(x)/(1 - 2*x*A(x)/(1 - 4*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 6*x*A(x)/(1 - ...))))))), a continued fraction.
0
1, 2, 12, 104, 1104, 13472, 183488, 2749056, 44996864, 802443776, 15579089920, 329170937856, 7562372632576, 188526816632832, 5083702487990272, 147676990509580288, 4600624321049722880, 153012055369679241216, 5409813656756850262016, 202534832564335070937088, 8001606648308588124045312
OFFSET
0,2
FORMULA
a(n) = [x^n] (Sum_{k>=0} A000165(k)*x^k)^(n+1)/(n + 1).
a(n) ~ sqrt(Pi) * (2*n)^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Nov 05 2021
EXAMPLE
G.f. A(x) = 1 + 2*x + 12*x^2 + 104*x^3 + 1104*x^4 + 13472*x^5 + 183488*x^6 + 2749056*x^7 + 44996864*x^8 + ...
log(A(x)) = 2*x + 20*x^2/2 + 248*x^3/3 + 3472*x^4/4 + 53152*x^5/5 + 878144*x^6/6 + ... + A293471(n)*x^n/n + ...
MATHEMATICA
Table[SeriesCoefficient[(1 + Sum[(2*k)!!*x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 27 2018
STATUS
approved