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URL: https://oeis.org/A382124

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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.
3
1, 1, 4, 9, 22, 44, 105, 200, 425, 825, 1634, 3072, 5917, 10846, 20153, 36436, 65882, 116831, 207293, 361502, 629539, 1083068, 1856251, 3150554, 5328137, 8933266, 14920357, 24745481, 40869317, 67089425, 109697089, 178379353, 288953043, 465805681, 748079686, 1196148976, 1905801579, 3024212984
OFFSET
0,3
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 root of A382125.
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
(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.
EXAMPLE
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 + ...
RELATED SERIES.
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 + ...
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = my(A = exp( sum(m=1, n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
Sequence in context: A105314 A200155 A381290 * A002835 A253289 A032288
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 06 2025
STATUS
approved