A185298 Expansion of e.g.f. x*exp(x)*exp(x*exp(x)).

0, 1, 4, 18, 92, 520, 3222, 21700, 157544, 1224576, 10133450, 88843084, 821832156, 7992373168, 81458868974, 867700216380, 9636146477648, 111323478770560, 1335253363581330, 16598183219157772, 213488758730421380, 2837046652845555696, 38899888173340835894

Offset: 0

Author

Geoffrey Critzer, Feb 20 2011

Keywords

• nonn

Comments

a(n) is the number of ways to designate an element in each block of the set partitions of {1,2,...,n} and then designate a block.

Inverse binomial transform: b(n) = Sum (-1)^(n-k)*C(n,k)*a(k), k=0..n of A052512. - Alexander R. Povolotsky, Oct 01 2011

Number of pointed set partitions of pointed sets k[1...k...n] for any point k. - Gus Wiseman, Sep 27 2015

Exponential series reversal gives A207833 with alternating signs: 1, -4, 30, -332, 4880, ... . - Vladimir Reshetnikov, Aug 04 2019

Examples

			The a(2) = 4 pointed set partitions are 1[1[12]], 1[1[1]2[2]], 2[1[1]2[2]], 2[2[12]].
The a(3) = 18 pointed set partitions are 1[1[123]], 1[1[1]2[23]], 1[1[1]3[23]], 1[1[12]3[3]], 1[1[13]2[2]], 1[1[1]2[2]3[3]], 2[2[123]], 2[1[1]2[23]], 2[1[13]2[2]], 2[2[2]3[13]], 2[2[12]3[3]], 2[1[1]2[2]3[3]], 3[3[123]], 3[1[1]3[23]], 3[1[12]3[3]], 3[2[2]3[13]], 3[2[12]3[3]], 3[1[1]2[2]3[3]].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..535
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A052512, A080108, A207833, A262671.

Programs

Mathematica

nn=30; a=x Exp[x]; Range[0,nn]! CoefficientList[Series[a Exp[a], {x,0,nn}],x]

PARI

x='x+O('x^33); concat([0],Vec(serlaplace(x*exp(x)*exp(x*exp(x))))) \\ Joerg Arndt, Oct 04 2015

Formula

E.g.f.: A(A(x)) where A(x) = x*exp(x).

a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-k+1). - Vladimir Kruchinin, Sep 23 2011

O.g.f.: Sum_{k>=1} k*x^k/(1 - k*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018

a(n) ~ exp(r*exp(r) + r - n) * n^(n + 1/2) / (r^(n - 1/2) * sqrt(1 + exp(r)*(1 + 3*r + r^2))), where r = 2*LambertW(exp(1/4)*sqrt(n)/2) - 1/2 + 1/(16*LambertW(exp(1/4)*sqrt(n)/2)^2 + LambertW(exp(1/4)*sqrt(n)/2) - 1). - Vaclav Kotesovec, Mar 21 2023