|
%I #22 Jul 10 2022 02:15:48
%S 1,1,3,12,62,390,2884,24472,234086,2490030,29139306,371878056,
%T 5138306700,76398336924,1215973642584,20624305367520,371309259462972,
%U 7071037633297116,141997246553420052,2998654325698019280,66426777891686458728,1540117294435707244488,37296711627004301923056
%N 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.
%F a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000085(k).
%F From _Emanuele Munarini_, Jul 09 2022: (Start)
%F a(n) = Sum_{k=0..n/2} |Stirling1(n+1,2*k+1)|*binomial(2*k,k)*k!/2^k.
%F a(n+1) = (n+1)*a(n) - Sum_{k=1..n} binomial(n,k)*(k-1)!*a(n-k). (End)
%p 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
%t nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Abs[StirlingS1[n, k]] HypergeometricU[-k/2, 1/2, -1/2]/(-1/2)^(k/2), {k, 0, n}], {n, 0, 22}]
%o (PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(log(1 - x)^2/2)/(1 - x))) \\ _Michel Marcus_, Jan 24 2019
%Y Cf. A000085, A004211, A008275, A130534, A319360.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 21 2019
|
