A235802 Expansion of e.g.f.: 1/(1 - x)^(2/(2-x)).
1, 1, 3, 12, 61, 375, 2697, 22176, 204977, 2102445, 23685615, 290642220, 3857751573, 55063797243, 840956549517, 13682498891040, 236257301424225, 4314883836968505, 83102361300891963, 1683252077760375660, 35770269996769203405, 795749735451309432255
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 96*x^3/3! + 976*x^4/4! + 12000*x^5/5! + ... where the logarithm involves sums of reciprocal binomial coefficients: log(A(x)) = x*(1) + x^2/2*(1 + 1) + x^3/3*(1 + 1/2 + 1) + x^4/4*(1 + 1/3 + 1/3 + 1) + x^5/5*(1 + 1/4 + 1/6 + 1/4 + 1) + x^6/6*(1 + 1/5 + 1/10 + 1/10 + 1/5 + 1) + ... Explicitly, the logarithm begins: log(A(x)) = x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 64*x^5/5! + 312*x^6/6! + 1812*x^7/7! + 12288*x^8/8! + ... + A003149(n-1)*x^n/n! + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..430
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( 1/(1-x)^(2/(2-x)) ))); // G. C. Greubel, Jul 12 2023 -
Mathematica
CoefficientList[Series[1/(1-x)^(2/(2-x)), {x,0,20}], x]*Range[0,20]! (* Vaclav Kotesovec, Jul 13 2014 *) -
PARI
{a(n)=n!*polcoeff(exp(sum(m=1, n, x^m/m*sum(k=0, m-1, 1/binomial(m-1, k))) +x*O(x^n)), n)} for(n=0,25,print1(a(n),", ")) -
PARI
{a(n)=n!*polcoeff(1/(1-x+x*O(x^n))^(2/(2-x)), n)} for(n=0,25,print1(a(n),", ")) -
SageMath
m=50 def f(x): return exp(sum(sum( 1/binomial(n-1,k) for k in range(n))*x^n/n for n in range(1,m+2))) def A235802_list(prec): P.
= PowerSeriesRing(QQ, prec) return P( f(x) ).egf_to_ogf().list() A235802_list(m) # G. C. Greubel, Jul 12 2023
Formula
E.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).
a(n) ~ n! * (n-2*log(n)). - Vaclav Kotesovec, Jul 13 2014