A206763 G.f.: Product_{n>=1} [ (1 - (-x)^n) / (1 - (n-1)^n*x^n) ]^(1/n).

1, 1, 0, 3, 23, 225, 2824, 42670, 762286, 15647321, 363901749, 9443387329, 270721307582, 8493470965716, 289518611494068, 10653599202688527, 420933469388468297, 17773313165985120798, 798686060913371460133, 38058408270727983373232

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..19.

Formula

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

Logarithmic derivative yields A206764.

Example

G.f.: A(x) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
where the g.f. equals the product:
A(x) = (1+x)/(1-0*x) * ((1-x^2)/(1-1^2*x^2))^(1/2) * ((1+x^3)/(1-2^3*x^3))^(1/3) * ((1-x^4)/(1-3^4*x^4))^(1/4) * ((1+x^5)/(1-4^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A206764:
log(A(x)) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...

Prog

(PARI) {a(n)=polcoeff(prod(k=1, n, ((1-(-1)^k*x^k)/(1-(k-1)^k*x^k +x*O(x^n)))^(1/k)), n)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*(-1)^(m-k))+x*O(x^n))), n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A206764 (log), A205814, A205811.

Sequence in context: A202997 A093162 A328808 * A306154 A201205 A068954

Adjacent sequences: A206760 A206761 A206762 * A206764 A206765 A206766

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved