A323630 Expansion of e.g.f. exp(log(1 - x)^2/2)/(1 - x). This is also the transform of the involution numbers given by the signless Stirling cycle numbers.
1, 1, 3, 12, 62, 390, 2884, 24472, 234086, 2490030, 29139306, 371878056, 5138306700, 76398336924, 1215973642584, 20624305367520, 371309259462972, 7071037633297116, 141997246553420052, 2998654325698019280, 66426777891686458728, 1540117294435707244488, 37296711627004301923056
Offset: 0
Keywords
Programs
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Maple
seq(n!*coeff(series(exp(log(1-x)^2/2)/(1-x),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019
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Mathematica
nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[Abs[StirlingS1[n, k]] HypergeometricU[-k/2, 1/2, -1/2]/(-1/2)^(k/2), {k, 0, n}], {n, 0, 22}] -
PARI
my(x='x + O('x^25)); Vec(serlaplace(exp(log(1 - x)^2/2)/(1 - x))) \\ Michel Marcus, Jan 24 2019
Formula
a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000085(k).
From Emanuele Munarini, Jul 09 2022: (Start)
a(n) = Sum_{k=0..n/2} |Stirling1(n+1,2*k+1)|*binomial(2*k,k)*k!/2^k.
a(n+1) = (n+1)*a(n) - Sum_{k=1..n} binomial(n,k)*(k-1)!*a(n-k). (End)