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.

1, 3, 15, 52, 180, 555, 1696, 4809, 13410, 35844, 93771, 238305, 594403, 1449441, 3476607, 8190824, 19015548, 43492230, 98197506, 218885763, 482337864, 1051051262, 2266904481, 4840955055, 10242621395, 21479302368, 44666897613, 92139573135, 188617118541, 383280793962, 773395096907

Offset: 0

Author

Paul D. Hanna, Apr 06 2025

Keywords

• nonn

Comments

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.

Equals the self-convolution cube of A382124.

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.

Examples

			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 + ...
where
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 + ...).
RELATED SERIES.
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A382124, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).

Cf. A087943, A329963.

Programs

Mathematica

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 *)

PARI

{a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).

(2) A(x) = exp( Sum_{n>=1} Sum_{k>=1} sigma(2*n*k) * x^(n*k) / n ).

(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k) * a(n-k) for n>0, with a(0) = 1.