A318694 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j)/(i*j)).

1, 1, 2, 10, 40, 248, 1868, 14516, 131920, 1409040, 15697872, 191687472, 2663239104, 37878672960, 582357866400, 9898540886880, 172534018584960, 3192686545714560, 63844374067107840, 1309775114921541120, 28512040933544970240, 656888836504576112640, 15495311684125737031680

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..447

Formula

E.g.f.: Product_{k>=1} (1 + x^k/k)^tau(k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-d)^(1-k/d)*tau(d) ) * x^k/k).

Maple

seq(n!*coeff(series(mul((1+x^k/k)^tau(k), k=1..100), x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j)/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k/k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-d)^(1 - k/d) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Crossrefs

Cf. A000005, A007838, A107742, A181541, A318416, A318693.

Sequence in context: A151028 A282619 A080252 * A281433 A277379 A227545

Adjacent sequences: A318691 A318692 A318693 * A318695 A318696 A318697

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 31 2018

Status

approved