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A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k). 6

%I

%S 1,1,3,8,50,94,2446,-9024,297216,-3183264,64191984,-1041792192,

%T 22098943632,-478805234064,11856288460272,-308662348027008,

%U 8575865689645440,-248582819381690880,7556655091130023680,-240521346554744194560,8049494171497089265920,-283469026458500121634560

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

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>

%F E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1 + x)^k/k).

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A000041(k)*k!.

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

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

%t nmax = 21; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[StirlingS1[n, k] PartitionsP[k] k!, {k, 0, n}], {n, 0, 21}]

%Y Cf. A000041, A006252, A053529, A167137, A320349.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Jun 17 2018

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Last modified June 18 03:46 EDT 2021. Contains 345098 sequences. (Running on oeis4.)