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%I #15 Jul 12 2023 05:52:38
%S 1,1,3,12,61,375,2697,22176,204977,2102445,23685615,290642220,
%T 3857751573,55063797243,840956549517,13682498891040,236257301424225,
%U 4314883836968505,83102361300891963,1683252077760375660,35770269996769203405,795749735451309432255
%N Expansion of e.g.f.: 1/(1 - x)^(2/(2-x)).
%H G. C. Greubel, <a href="/A235802/b235802.txt">Table of n, a(n) for n = 0..430</a>
%F E.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).
%F E.g.f.: exp( Sum_{n>=1} A003149(n-1)*x^n/n! ), where A003149(n) = Sum_{k=0..n} k!*(n-k)!.
%F a(n) ~ n! * (n-2*log(n)). - _Vaclav Kotesovec_, Jul 13 2014
%e E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 96*x^3/3! + 976*x^4/4! + 12000*x^5/5! + ...
%e where the logarithm involves sums of reciprocal binomial coefficients:
%e 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) + ...
%e Explicitly, the logarithm begins:
%e 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! + ...
%t CoefficientList[Series[1/(1-x)^(2/(2-x)), {x,0,20}], x]*Range[0,20]! (* _Vaclav Kotesovec_, Jul 13 2014 *)
%o (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)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) {a(n)=n!*polcoeff(1/(1-x+x*O(x^n))^(2/(2-x)), n)}
%o for(n=0,25,print1(a(n),", "))
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( 1/(1-x)^(2/(2-x)) ))); // _G. C. Greubel_, Jul 12 2023
%o (SageMath)
%o m=50
%o 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)))
%o def A235802_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(x) ).egf_to_ogf().list()
%o A235802_list(m) # _G. C. Greubel_, Jul 12 2023
%Y Cf. A003149, A193425.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 15 2014
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