A305199 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).

1, 2, 6, 28, 152, 1008, 7756, 67688, 659424, 7123776, 84154224, 1079913888, 14962632384, 222447507072, 3531920599008, 59664827178048, 1067975819206656, 20192760528611328, 402169396496004864, 8414121277765679616, 184498963978904644608, 4231186653661629843456

Offset: 0

Author

Ilya Gutkovskiy, May 27 2018

Keywords

• nonn

Comments

Exponential convolution of the sequences A007838 and A007841.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A007838, A007841, A292358, A292359, A295792.

Programs

Maple

a:=series(mul((1+x^k/k)/(1-x^k/k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)/(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

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

a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / exp(n + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 26 2019