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A156304
G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.
5
1, 1, 8, 21, 77, 199, 661, 1663, 4852, 12382, 33289, 82877, 213026, 518109, 1279852, 3053404, 7312985, 17093793, 39952528, 91661695, 209709116, 473095589, 1062567288, 2359804486, 5214774263, 11415904502, 24860918943, 53709881911
OFFSET
0,3
COMMENTS
Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Euler transform of A160889. - Vaclav Kotesovec, Nov 01 2024
LINKS
FORMULA
a(n) = (1/n)*Sum_{k=1..n} sigma(k^3) * a(n-k) for n>0, with a(0)=1.
log(a(n)) ~ 4*Pi*c^(1/4)*n^(3/4) / (3^(5/4)*5^(1/4)), where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699... - Vaclav Kotesovec, Nov 01 2024
EXAMPLE
G.f.: A(x) = 1 + x + 8*x^2 + 21*x^3 + 77*x^4 + 199*x^5 + 661*x^6 +...
log(A(x)) = x + 15*x^2/2 + 40*x^3/3 + 127*x^4/4 + 156*x^5/5 + 600*x^6/6 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^3)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^3)*a(n-k)))}
CROSSREFS
Cf. A000203 (sigma), A000041 (partitions), A156303, A202993, A203557.
Sequence in context: A267144 A240516 A079386 * A232049 A225658 A226313
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Feb 08 2009
STATUS
approved