A206765 G.f.: Product_{n>=1} [ (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ]^(1/n).

1, 1, 12, 87, 907, 8393, 118932, 1683990, 31209334, 635005549, 15054451057, 393600573337, 11466736952722, 363842430190308, 12564913404375244, 467483278911401155, 18670853023655302285, 795978439482823960066, 36093307429580735618893

Offset

0,3

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k) ).

Logarithmic derivative yields A206766.

a(n) ~ exp(3) * n^(n-1). - Vaclav Kotesovec, Oct 08 2016

Example

G.f.: A(x) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
where the g.f. equals the product:
A(x) = (1-3*x)/(1-4*x) * ((1-3^2*x^2)/(1-5^2*x^2))^(1/2) * ((1-3^3*x^3)/(1-6^3*x^3))^(1/3) * ((1-3^4*x^4)/(1-7^4*x^4))^(1/4) * ((1-3^5*x^5)/(1-8^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A206766:
log(A(x)) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...

Mathematica

max = 19; p = Product[((1-3^n*x^n) / (1-(n+3)^n*x^n))^(1/n), {n, 1, max}] + O[x]^max; CoefficientList[p, x] (* Jean-François Alcover, Oct 08 2016 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*3^(m-k))+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(prod(k=1, n, ((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))^(1/k)), n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A206766 (log), A205814, A205811, A206763.

Sequence in context: A183721 A180797 A137207 * A228500 A348415 A082814

Adjacent sequences: A206762 A206763 A206764 * A206766 A206767 A206768

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved