A174465 G.f.: exp( Sum_{n>=1} A174466(n)*x^n/n ) where A174466(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

1, 1, 4, 7, 19, 31, 74, 122, 258, 430, 835, 1378, 2557, 4162, 7382, 11932, 20471, 32676, 54634, 86251, 141001, 220371, 353413, 546783, 863043, 1322425, 2057525, 3125092, 4801297, 7230393, 10984924, 16410474, 24679719, 36593278, 54526145, 80272501

Offset: 0

Author

Paul D. Hanna, Apr 04 2010

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

Programs

Mathematica

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)}

Formula

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