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A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

%I A382124 #8 Apr 06 2025 14:55:22
%S A382124 1,1,4,9,22,44,105,200,425,825,1634,3072,5917,10846,20153,36436,65882,
%T A382124 116831,207293,361502,629539,1083068,1856251,3150554,5328137,8933266,
%U A382124 14920357,24745481,40869317,67089425,109697089,178379353,288953043,465805681,748079686,1196148976,1905801579,3024212984
%N A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
%C A382124 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 A382124 Equals the self-convolution cube root of A382125.
%C A382124 Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.
%H A382124 Paul D. Hanna, <a href="/A382124/b382124.txt">Table of n, a(n) for n = 0..1000</a>
%F A382124 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A382124 (1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
%F A382124 (2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
%F A382124 (3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k)/3 * a(n-k) for n > 0, with a(0) = 1.
%e A382124 G.f.: A(x) = 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 + 3072*x^11 + 5917*x^12 + ...
%e A382124 RELATED SERIES.
%e A382124 A(x)^3 = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + ... + A382125(n)*x^n + ...
%t A382124 nmax=37; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/(3n) ,{n,nmax}]],{x,0,nmax}],x] (* _Stefano Spezia_, Apr 06 2025 *)
%o A382124 (PARI) {a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
%o A382124 polcoef(A,n)}
%o A382124 for(n=0,30, print1(a(n),", "))
%Y A382124 Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
%Y A382124 Cf. A087943, A329963.
%K A382124 nonn
%O A382124 0,3
%A A382124 _Paul D. Hanna_, Apr 06 2025