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A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

%I A382125 #10 Apr 06 2025 14:55:34
%S A382125 1,3,15,52,180,555,1696,4809,13410,35844,93771,238305,594403,1449441,
%T A382125 3476607,8190824,19015548,43492230,98197506,218885763,482337864,
%U A382125 1051051262,2266904481,4840955055,10242621395,21479302368,44666897613,92139573135,188617118541,383280793962,773395096907
%N A382125 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
%C A382125 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.
%C A382125 Equals the self-convolution cube of A382124.
%C A382125 Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.
%H A382125 Paul D. Hanna, <a href="/A382125/b382125.txt">Table of n, a(n) for n = 0..1000</a>
%F A382125 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A382125 (1) A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
%F A382125 (2) A(x) = exp( Sum_{n>=1} Sum_{k>=1} sigma(2*n*k) * x^(n*k) / n ).
%F A382125 (3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k) * a(n-k) for n>0, with a(0) = 1.
%e A382125 G.f.: A(x) = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + 35844*x^9 + 93771*x^10 + ...
%e A382125 where
%e A382125 A(x) = exp(3*x + 21*x^2/2 + 48*x^3/3 + 105*x^4/4 + 108*x^5/5 + 336*x^6/6 + 192*x^7/7 + 465*x^8/8 + 507*x^9/9 + 756*x^10/10 + ... + sigma(n)*sigma(2*n)*x^n/n + ...).
%e A382125 RELATED SERIES.
%e A382125 A(x)^(1/3) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + ... + A382124(n)*x^n + ...
%t A382125 nmax=30; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/n ,{n,nmax}]],{x,0,nmax}],x] (* _Stefano Spezia_, Apr 06 2025 *)
%o A382125 (PARI) {a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)*x^m/m ) +x*O(x^n) ));
%o A382125 polcoef(A,n)}
%o A382125 for(n=0,30, print1(a(n),", "))
%Y A382125 Cf. A382124, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
%Y A382125 Cf. A087943, A329963.
%K A382125 nonn
%O A382125 0,2
%A A382125 _Paul D. Hanna_, Apr 06 2025