VOOZH about

URL: https://oeis.org/A382125

⇱ A382125 - OEIS

login
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.
4
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
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).
Sequence in context: A192742 A166035 A038192 * A212869 A371481 A332375
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 06 2025
STATUS
approved