A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

1, 1, 14, 35, 205, 521, 2507, 6709, 26712, 73834, 262431, 724537, 2384988, 6552033, 20289864, 55244988, 163342701, 439201501, 1251532060, 3321188863, 9177476977, 24028568664, 64709650590, 167153761523, 440300702427, 1122562426240, 2900254892900, 7301575351055, 18544013542057

Offset

0,3

Comments

Compare with g.f. for partition numbers A000041: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.

Links

Table of n, a(n) for n=0..28.

Formula

a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.

G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [Paul D. Hanna, Jan 31 2012]

Example

G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
such that, by definition,
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 6^3*x^5/5 + 12^3*x^6/6 +...

Prog

(PARI) N=100; v=Vec(exp(sum(k=1, N, sigma(k)^3*x^k/k)+x*O(x^N)))
(PARI) a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^2*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A156302, A205797.

Sequence in context: A321135 A330207 A104317 * A203803 A115664 A182753

Adjacent sequences: A178930 A178931 A178932 * A178934 A178935 A178936

Keyword

nonn

Author

Joerg Arndt, Dec 30 2010

Status

approved