A224439 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^(n-1) * x^n/n ).

1, 1, 2, 7, 93, 357, 41927, 80065, 21483964, 112388242, 19973468103, 25813956365, 691174602929572, 695655501206181, 63995738768530056, 1469847380380956056, 1468171845473348201557, 1477216529008886240457, 62064992121198579569054696, 62086294811417506896412871

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.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..335

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^(k-1) * a(n-k) for n > 0, with a(0)=1.

Logarithmic derivative yields A224440.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 93*x^4 + 357*x^5 + 41927*x^6 + ... where
log(A(x)) = x + 3^1*x^2/2 + 4^2*x^3/3 + 7^3*x^4/4 + 6^4*x^5/5 + 12^5*x^6/6 + ...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m)^(m-1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^(k-1)*a(n-k)))}

Crossrefs

Cf. A224440, A156217, A158095, A000203.

Sequence in context: A096208 A123995 A350754 * A304722 A056161 A076740

Adjacent sequences: A224436 A224437 A224438 * A224440 A224441 A224442

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2013

Status

approved