cp's OEIS Frontend

A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

%I A143405 #66 Jul 22 2022 04:03:23
%S A143405 1,1,4,17,89,552,3895,30641,265186,2497551,25373097,276105106,
%T A143405 3199697517,39297401197,509370849148,6943232742493,99217486649933,
%U A143405 1482237515573624,23093484367004715,374416757914118941,6304680593346141746,110063311977033807187
%N A143405 Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.
%C A143405 a(n) is the number of the partitions of an n-set where each block is endowed with a nonempty subset. - _Emanuele Munarini_, Sep 15 2016
%H A143405 Alois P. Heinz, <a href="/A143405/b143405.txt">Table of n, a(n) for n = 0..504</a>
%H A143405 Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022.
%H A143405 Vaclav Kotesovec, <a href="/A143405/a143405.pdf">Asymptotics of OEIS A143405, A355291 and the generalization</a>, Jul 12 2022 (this version also includes several figures).
%H A143405 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A143405 a(n) = Sum_{k=0..n} Sum_{t=k..n} C(n,t) * Stirling2(t,k)*k^(n-t).
%F A143405 a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k) * Stirling2(k,t)*t^(n-k).
%F A143405 a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t)*t^(k-t).
%F A143405 E.g.f.: exp(exp(x)*(exp(x)-1)). - _Vladeta Jovovic_, Dec 08 2008
%F A143405 a(n) = sum(binomial(n,k)*2^k*bell(k)*S(n-k,-1),k=0..n), where bell(n) are the Bell numbers (A000110) and S(n,x) = sum(Stirling2(n,k)*x^k,k=0..n) are the Stirling (or exponential) polynomials. - _Emanuele Munarini_, Sep 15 2016
%F A143405 Identity: sum(binomial(n,k)*a(k)*bell(n-k),k=0..n) = 2^n*bell(n). - _Emanuele Munarini_, Sep 15 2016
%F A143405 a(n) = Sum_{k=0..n} A047974(k) * Stirling2(n,k). - _Seiichi Manyama_, May 14 2022
%F A143405 a(n) ~ exp(exp(2*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) - (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) - 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - _Vaclav Kotesovec_, Jul 03 2022
%F A143405 a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 1/8 - n/LambertW(n) + sqrt(n/LambertW(n)))). - _Vaclav Kotesovec_, Jul 08 2022
%e A143405 a(2) = 4, because there are 4 forests for 2 labels: {1,2}, {1}{2}, {1}<-2, {2}<-1.
%p A143405 a:= n-> add(add(binomial(n, t)*Stirling2(t, k)*k^(n-t), t=k..n), k=0..n):
%p A143405 seq(a(n), n=0..30);
%p A143405 # second Maple program:
%p A143405 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A143405       a(n-j)*binomial(n-1, j-1)*(2^j-1), j=1..n))
%p A143405     end:
%p A143405 seq(a(n), n=0..23);  # _Alois P. Heinz_, Oct 05 2019
%t A143405 CoefficientList[Series[Exp[Exp[t] (Exp[t] - 1)], {t, 0, 12}], t] Range[0, 12]! (* _Emanuele Munarini_, Sep 15 2016 *)
%t A143405 Table[Sum[Binomial[n, k] 2^k BellB[k] BellB[n - k, -1], {k, 0, n}], {n, 0, 12}] (* _Emanuele Munarini_, Sep 15 2016 *)
%t A143405 Table[Sum[BellY[n, k, 2^Range[n] - 1], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A143405 (PARI) a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*stirling(n, k, 2)); \\ _Seiichi Manyama_, May 14 2022
%Y A143405 Cf. A055882, A143395, A143396, A143397, A048993, A008277, A007318, A000110, A355291.
%K A143405 nonn
%O A143405 0,3
%A A143405 _Alois P. Heinz_, Aug 12 2008