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%I #6 Mar 27 2019 03:54:00
%S 1,2,6,26,126,742,4986,37942,321502,3026150,31198206,351179182,
%T 4282131354,56334933358,795191463982,12001157392246,192825757504222,
%U 3288240179785318,59314678786251486,1128751491248706814,22599321692994969886,474961934284902165190,10454818842695667265942
%N Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)/(1 - x^k/k!).
%C Exponential convolution of the sequences A005651 and A007837.
%F E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (1 + (-1)^(k+1))*x^(j*k)/(k*(j!)^k)).
%p 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
%t nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t 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]!
%Y Cf. A005651, A007837, A305199.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 17 2018
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