A198301 G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{d|n} d*sigma(n/d,d) ).

1, 1, 3, 5, 12, 18, 42, 62, 131, 206, 398, 610, 1203, 1810, 3358, 5260, 9471, 14518, 26182, 39906, 70320, 108849, 187251, 287525, 497288, 758860, 1286936, 1986352, 3330677, 5102712, 8560107, 13070327, 21685731, 33328561, 54744685, 83792111, 137817745, 210223967

Offset

0,3

Comments

Here sigma(n,k) is the sum of the k-th powers of the divisors of n.

Links

Table of n, a(n) for n=0..37.

Formula

G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n ).

Logarithmic derivative yields A198302.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 42*x^6 + 62*x^7 +...
where the logarithm begins:
log(A(x)) = x + 5*x^2/2 + 7*x^3/3 + 21*x^4/4 + 11*x^5/5 + 65*x^6/6 + 15*x^7/7 + 133*x^8/8 + 106*x^9/9 +...+ A198302(n)*x^n/n +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, d*sigma(m/d, d))*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m, k)*x^(m*k)/m)+x*O(x^n))), n)}

Crossrefs

Cf. A198302 (log), A198296.

Sequence in context: A197988 A025088 A279784 * A323866 A082740 A010067

Adjacent sequences: A198298 A198299 A198300 * A198302 A198303 A198304

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2012

Status

approved