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

0,3

Links

Table of n, a(n) for n=0..22.

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)

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

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}]

Prog

(PARI) my(x='x + O('x^25)); Vec(serlaplace(exp(log(1 - x)^2/2)/(1 - x))) \\ Michel Marcus, Jan 24 2019

Crossrefs

Cf. A000085, A004211, A008275, A130534, A319360.

Sequence in context: A187820 A074529 A143916 * A020033 A266329 A208734

Adjacent sequences: A323627 A323628 A323629 * A323631 A323632 A323633

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 21 2019

Status

approved