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A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.

%I A156302 #25 Apr 06 2025 15:21:03
%S A156302 1,1,5,10,30,57,152,289,676,1304,2809,5335,10961,20487,40329,74476,
%T A156302 141914,258094,479638,860025,1563716,2767982,4940567,8636563,15173805,
%U A156302 26217392,45416811,77629455,132800937,224695510,380079521,637006921
%N A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.
%C A156302 Compare with g.f. for 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 A156302 Vaclav Kotesovec, <a href="/A156302/b156302.txt">Table of n, a(n) for n = 0..10000</a>
%F A156302 a(n) = (1/n)*Sum_{k=1..n} sigma(k)^2*a(n-k) for n>0, with a(0) = 1.
%F A156302 Euler transform of A060648. [From _Vladeta Jovovic_, Feb 14 2009]
%F A156302 It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From _Joerg Arndt_, Dec 30 2010]
%F A156302 G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From _Paul D. Hanna_, Jan 23 2012]
%F A156302 log(a(n)) ~ 3*(5*zeta(3))^(1/3) * n^(2/3) / 2. - _Vaclav Kotesovec_, Oct 29 2024
%e A156302 G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...
%e A156302 log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...
%e A156302 Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +
%e A156302 (3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +
%e A156302 (4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +
%e A156302 (7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +
%e A156302 (6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...
%t A156302 nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 29 2024 *)
%o A156302 (PARI) {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)^2*x^k/k)+x*O(x^n)),n)}
%o A156302 (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^2*a(n-k)))}
%o A156302 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,sum(k=1,n\m,sigma(m*k)*x^(m*k)/m)+x*O(x^n))),n)}
%Y A156302 Cf. A000203 (sigma), A000041 (partitions), A072861, A178933, A205797, A382125.
%K A156302 nonn
%O A156302 0,3
%A A156302 _Paul D. Hanna_, Feb 08 2009