0,3

Compare to the g.f. of the number of planar partitions of n (A000219):

exp( Sum_{n>=1} sigma_2(n)*x^n/n ) where sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d).

tau(n) = A000005(n) = the number of divisors of n,

and sigma(n) = A000203(n) = sum of divisors of n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

G.f.: Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k)). - Vaclav Kotesovec, Jan 04 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_3(k), where tau_3() = A007425. - Ilya Gutkovskiy, May 22 2018

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)

nmax = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A007425[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2018 *)

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, x^m/m*sumdiv(m, d, d*sigma(m/d)*sigma(d, 0)))+x*O(x^n)), n)}

Cf. A174466, A000203 (sigma), A000005 (tau), A006171, A007425, A280473, A280487.

Sequence in context: A063605 A024824 A164265 * A006381 A318099 A274691

Adjacent sequences: A174462 A174463 A174464 * A174466 A174467 A174468

Paul D. Hanna, Apr 04 2010

approved