A185298 - OEIS

%I #51 Mar 05 2024 11:26:38

%S 0,1,4,18,92,520,3222,21700,157544,1224576,10133450,88843084,

%T 821832156,7992373168,81458868974,867700216380,9636146477648,

%U 111323478770560,1335253363581330,16598183219157772,213488758730421380,2837046652845555696,38899888173340835894

%N Expansion of e.g.f. x*exp(x)*exp(x*exp(x)).

%C 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.

%C 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

%C Number of pointed set partitions of pointed sets k[1...k...n] for any point k. - _Gus Wiseman_, Sep 27 2015

%C Exponential series reversal gives A207833 with alternating signs: 1, -4, 30, -332, 4880, ... . - _Vladimir Reshetnikov_, Aug 04 2019

%H G. C. Greubel, <a href="/A185298/b185298.txt">Table of n, a(n) for n = 0..535</a>

%H Vaclav Kotesovec, <a href="/A185298/a185298.jpg">Graph - the asymptotic ratio (10000 terms)</a>

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

%F a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-k+1). - _Vladimir Kruchinin_, Sep 23 2011

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

%F 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

%e The a(2) = 4 pointed set partitions are 1[1[12]], 1[1[1]2[2]], 2[1[1]2[2]], 2[2[12]].

%e 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]].

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

%o (PARI) x='x+O('x^33); concat([0],Vec(serlaplace(x*exp(x)*exp(x*exp(x))))) \\ _Joerg Arndt_, Oct 04 2015

%Y Cf. A052512, A080108, A207833, A262671.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Feb 20 2011