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%I #7 Oct 26 2017 22:58:18
%S 1,1,2,6,25,123,695,4710,38573,363290,3762248,42712427,544108640,
%T 7858522102,125629094752,2132933782538,37180929264503,653233840242274,
%U 11489698066482812,202366696882463928,3583045708114956712,64151951314880747470,1170351942994473380196,21993337390128064959103,433693240259690293897796,9276431005281628486906887,226051070328728855263836057
%N O.g.f.: exp( Sum_{n>=1} A251658(n) * x^n/n ), where A251658(n) = Sum_{k=1..n} C(n,k) * sigma(k)^(n-k).
%H Paul D. Hanna, <a href="/A291558/b291558.txt">Table of n, a(n) for n = 0..200</a>
%F O.g.f.: exp( Sum_{n>=1} x^n/(1 - sigma(n)*x)^n / n ). - _Paul D. Hanna_, Oct 26 2017
%e O.g.f: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 123*x^5 + 695*x^6 + 4710*x^7 + 38573*x^8 + 363290*x^9 + 3762248*x^10 + 42712427*x^11 + 544108640*x^12 + 7858522102*x^13 + 125629094752*x^14 + 2132933782538*x^15 +...
%e such that
%e log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 75*x^4/4 + 471*x^5/5 + 3273*x^6/6 + 26916*x^7/7 + 262027*x^8/8 + 2851375*x^9/9 +...+ A251658(n)*x^n/n +...
%e Also,
%e log(A(x)) = x/(1-x) + x^2/(1-3*x)^2/2 + x^3/(1-4*x)^3/3 + x^4/(1-7*x)^4/4 + x^5/(1-6*x)^5/5 + x^6/(1-12*x)^6/6 + x^7/(1-8*x)^7/7 + x^8/(1-15*x)^8/8 + x^9/(1-13*x)^9/9 + x^10/(1-18*x)^10/10 +...+ x^n/(1-sigma(n)*x)^n/n +...
%o (PARI) {A251658(n)=sum(k=1, n, binomial(n, k)*sigma(k)^(n-k))}
%o {a(n) = polcoeff( exp( sum(m=1,n+1, A251658(m)*x^m/m) +x*O(x^n) ),n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = polcoeff( exp( sum(m=1,n+1, x^m/(1-sigma(m)*x+x*O(x^n))^m/m) ),n)}
%o for(n=0, 30, print1(a(n), ", ")) \\ - _Paul D. Hanna_, Oct 26 2017
%Y Cf. A251658.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 26 2017
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