A355291 -id:A355291 - 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.

1, 1, 4, 17, 89, 552, 3895, 30641, 265186, 2497551, 25373097, 276105106, 3199697517, 39297401197, 509370849148, 6943232742493, 99217486649933, 1482237515573624, 23093484367004715, 374416757914118941, 6304680593346141746, 110063311977033807187

Offset: 0

Alois P. Heinz, Aug 12 2008

nonn

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

			a(2) = 4, because there are 4 forests for 2 labels: {1,2}, {1}{2}, {1}<-2, {2}<-1.

Alois P. Heinz, Table of n, a(n) for n = 0..504
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).
Index entries for sequences related to rooted trees

Cf. A055882, A143395, A143396, A143397, A048993, A008277, A007318, A000110, A355291.

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

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 *)

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

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

A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)).

Original entry on oeis.org

1, 2, 12, 82, 688, 6754, 75096, 928386, 12591392, 185384130, 2938319144, 49799613538, 897495547184, 17118975292514, 344206910941624, 7270287035936706, 160826794265399360, 3716047107259486082, 89472755268582494792, 2240097688067896960674, 58207872357772581544272

Offset: 0

Vaclav Kotesovec, Jun 30 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..467

Cf. A143405, A355291, A355379, A355381.

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(x)))) \\ Michel Marcus, Jun 30 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k, -1).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) - 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) - (n/LambertW(n))^(1/3) - n) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022

A355381 Expansion of e.g.f. exp(exp(3x) - exp(2x)).

Original entry on oeis.org

1, 1, 6, 35, 247, 2102, 20547, 224541, 2707292, 35638329, 507464939, 7757439428, 126538995293, 2191454313661, 40120212534838, 773554002955047, 15656660861190371, 331700076893737054, 7337160433117899959, 169068422994937678185, 4050093664805130165348

Offset: 0

Vaclav Kotesovec, Jun 30 2022

nonn

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022

In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..471

Cf. A143405, A194006, A355291, A355380, A355378.

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[2*x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * 2^(n-k) * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(2*x)))) \\ Michel Marcus, Jun 30 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * 2^(n-k) * Bell(k) * Bell(n-k, -1).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) - exp(2*z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - 2*(1 + 2*z)*exp(2*z)), where z = LambertW(n)/3 - 1/(2 + 3/LambertW(n) - 9 * n^(1/3) * (1 + LambertW(n)) / (2*LambertW(n)^(4/3))). - Vaclav Kotesovec, Jul 03 2022

A355380 Expansion of e.g.f. exp(exp(3x) + exp(2x) - 2).

Original entry on oeis.org

1, 5, 38, 355, 3879, 48050, 661163, 9961745, 162598044, 2851150665, 53350521523, 1059447004560, 22224898346989, 490589320542305, 11356591577861398, 274886065370874775, 6939205217774546339, 182273695066097752170, 4971724931587003394863, 140559648864263508395965

Offset: 0

Vaclav Kotesovec, Jun 30 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..463

Cf. A143405, A355291, A355381.

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[2*x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * 2^(n-k) * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(2*x) - 2))) \\ Michel Marcus, Jun 30 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * 2^(n-k) * Bell(k) * Bell(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) + exp(2*z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) + 2*(1 + 2*z)*exp(2*z)), where z = LambertW(n)/3 - 1/(2 + 3/LambertW(n) + 9 * n^(1/3) * (1 + LambertW(n)) / (2*LambertW(n)^(4/3))). - Vaclav Kotesovec, Jul 03 2022

A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).

Original entry on oeis.org

1, 4, 26, 212, 2046, 22588, 278942, 3792916, 56128254, 895795692, 15307847614, 278435732484, 5364073445278, 108994074306268, 2327475127169182, 52069279762495220, 1217024509006768574, 29647115491635327180, 751085909757123127294, 19750410883486281805028

Offset: 0

Vaclav Kotesovec, Jun 30 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..466

Cf. A143405, A355291, A355378.

nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(x) - 2))) \\ Michel Marcus, Jun 30 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022
a(n) ~ exp(exp(3*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) + (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) + 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) + (n/LambertW(n))^(1/3) - n - 2) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022

A355423 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 6, 14, 5, 0, 1, 10, 50, 81, 15, 0, 1, 15, 130, 504, 551, 52, 0, 1, 21, 280, 2000, 5870, 4266, 203, 0, 1, 28, 532, 6075, 35054, 76872, 36803, 877, 0, 1, 36, 924, 15435, 148429, 684000, 1111646, 348543, 4140, 0

Offset: 0

Seiichi Manyama, Jul 01 2022

			Square array begins:
  1,  1,    1,     1,      1,       1, ...
  0,  1,    3,     6,     10,      15, ...
  0,  2,   14,    50,    130,     280, ...
  0,  5,   81,   504,   2000,    6075, ...
  0, 15,  551,  5870,  35054,  148429, ...
  0, 52, 4266, 76872, 684000, 4004100, ...

Columns k=0-4 give: A000007, A000110, A355291, A355421, A355422.
Main diagonal gives A320288.
Cf. A306024, A320253.

T(0,k) = 1 and T(n,k) = Sum_{i=1..n} (Sum_{j=1..k} j^i) * binomial(n-1,i-1) * T(n-i,k) for n > 0.

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.

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A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)).

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A355381 Expansion of e.g.f. exp(exp(3*x) - exp(2*x)).

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A355380 Expansion of e.g.f. exp(exp(3*x) + exp(2*x) - 2).

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Mathematica

PARI

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A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A355423 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).

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Examples

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A355381 Expansion of e.g.f. exp(exp(3x) - exp(2x)).

A355380 Expansion of e.g.f. exp(exp(3x) + exp(2x) - 2).