A318694 -id:A318694 - cp's OEIS frontend

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij) / (ij)!).

1, 1, 2, 8, 17, 87, 366, 1514, 8770, 45585, 267586, 1612624, 11914416, 73215391, 522906754, 4364545708, 33150679697, 263662491935, 2151338992440, 20815916251604, 178593028936507, 1714283809331191, 15531842607259512, 158682350653110712, 1667852117293837230

Offset: 0

Ilya Gutkovskiy, Feb 13 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..610

Cf. A000005, A007837, A032315, A107742, A318694, A340903, A341505, A341876.

nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, -(n - 1)! Sum[Sum[d DivisorSigma[0, d]/(-d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

E.g.f.: Product_{k>=1} (1 + x^k / k!)^sigma_0(k).

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij)/(ij)).

Original entry on oeis.org

1, 1, 4, 16, 100, 628, 5388, 46212, 491328, 5381760, 68023056, 892073136, 13238778144, 201822014496, 3397195558560, 59356290115296, 1121097742183296, 21916440531679104, 459855848691876096, 9952944631606759680, 229191463614349301760, 5446997871156332605440, 136439919208493792455680

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Cf. A000005, A006171, A007841, A294469, A318415, A318694.

seq(n!*coeff(series(mul(mul(1/(1-x^(i*j)/(i*j)),i=1..30),j=1..30),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[1/(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/(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}]

E.g.f.: Product_{k>=1} 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).

cp's OEIS Frontend

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij) / (ij)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij)/(ij)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(i*j) / (i*j)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j)/(i*j)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(ij) / (ij)!).

A318693 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij)/(ij)).