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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.
13
1, 1, 4, 17, 89, 552, 3895, 30641, 265186, 2497551, 25373097, 276105106, 3199697517, 39297401197, 509370849148, 6943232742493, 99217486649933, 1482237515573624, 23093484367004715, 374416757914118941, 6304680593346141746, 110063311977033807187
OFFSET
0,3
COMMENTS
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
LINKS
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vaclav Kotesovec, Asymptotics of OEIS A143405, A355291 and the generalization, Jul 12 2022 (this version also includes several figures).
FORMULA
a(n) = Sum_{k=0..n} Sum_{t=k..n} C(n,t) * Stirling2(t,k)*k^(n-t).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k) * Stirling2(k,t)*t^(n-k).
a(n) = Sum_{k=0..n} Sum_{t=0..k} C(n,k-t) * Stirling2(n-(k-t),t)*t^(k-t).
E.g.f.: exp(exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
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
Identity: sum(binomial(n,k)*a(k)*bell(n-k),k=0..n) = 2^n*bell(n). - Emanuele Munarini, Sep 15 2016
a(n) = Sum_{k=0..n} A047974(k) * Stirling2(n,k). - Seiichi Manyama, May 14 2022
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
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
EXAMPLE
a(2) = 4, because there are 4 forests for 2 labels: {1,2}, {1}{2}, {1}<-2, {2}<-1.
MAPLE
a:= n-> add(add(binomial(n, t)*Stirling2(t, k)*k^(n-t), t=k..n), k=0..n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-j)*binomial(n-1, j-1)*(2^j-1), j=1..n))
end:
seq(a(n), n=0..23); # Alois P. Heinz, Oct 05 2019
MATHEMATICA
CoefficientList[Series[Exp[Exp[t] (Exp[t] - 1)], {t, 0, 12}], t] Range[0, 12]! (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[Binomial[n, k] 2^k BellB[k] BellB[n - k, -1], {k, 0, n}], {n, 0, 12}] (* Emanuele Munarini, Sep 15 2016 *)
Table[Sum[BellY[n, k, 2^Range[n] - 1], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
PROG
(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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 12 2008
STATUS
approved