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

0,2

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.

Links

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

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.

Example

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 + ...

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 *)

Prog

(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), ", "))

Crossrefs

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

Cf. A087943, A329963.

Sequence in context: A192742 A166035 A038192 * A212869 A371481 A332375

Adjacent sequences: A382122 A382123 A382124 * A382126 A382127 A382128

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2025

Status

approved