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A156217 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.

%I A156217 #14 Nov 11 2017 04:53:18
%S A156217 1,1,5,26,634,2273,502568,821149,323391480,1514316108,360153555251,
%T A156217 440146271717,19353735645240631,19423959923754863,1599762850473552085,
%U A156217 35664862808194240282,45517618403498070780338,45844669861151626268272
%N A156217 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.
%C A156217 Logarithmic derivative yields A217872.
%C A156217 Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
%H A156217 Seiichi Manyama, <a href="/A156217/b156217.txt">Table of n, a(n) for n = 0..335</a>
%F A156217 a(n) = (1/n)*Sum_{k=1..n} sigma(k)^k * a(n-k) for n>0, with a(0)=1.
%e A156217 G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 + ...
%e A156217 log(A(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 + ...
%p A156217 A156217 := proc(n) option remember ; if n = 0 then 1; else add( (numtheory[sigma](k))^k*procname(n-k),k=1..n)/n ; fi; end: seq(A156217(n),n=0..10) ; # _R. J. Mathar_, Apr 02 2009
%o A156217 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(m)^m*x^m/m)+x*O(x^n)),n)}
%o A156217 (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^k*a(n-k)))}
%Y A156217 Cf. A217872, A224439, A000041, A000203.
%K A156217 nonn
%O A156217 0,3
%A A156217 _Paul D. Hanna_, Feb 06 2009