A306041 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)/(1 - x^k/k!).

1, 2, 6, 26, 126, 742, 4986, 37942, 321502, 3026150, 31198206, 351179182, 4282131354, 56334933358, 795191463982, 12001157392246, 192825757504222, 3288240179785318, 59314678786251486, 1128751491248706814, 22599321692994969886, 474961934284902165190, 10454818842695667265942

Offset: 0

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

• nonn

Comments

Exponential convolution of the sequences A005651 and A007837.

Crossrefs

Cf. A005651, A007837, A305199.

Programs

Maple

a:=series(mul((1+x^k/k!)/(1-x^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; 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)).