A143396 -id:A143396 - 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

A273652 Number of forests of labeled rooted trees of height at most 1, with n labels, two of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

2, 9, 30, 90, 255, 693, 1820, 4644, 11565, 28215, 67650, 159822, 372827, 860265, 1966200, 4456584, 10027161, 22413483, 49807550, 110100690, 242221287, 530579709, 1157628180, 2516582700, 5452595525, 11777605983, 25367150970, 54492397974, 116769423795

Offset: 2

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A143396.

a:= n-> binomial(n,2)*add(Stirling2(2,j)*j^(n-2), j=0..2):
seq(a(n), n=2..40);

E.g.f.: x^2/2! * Sum_{j=0..2} Stirling2(2,j)*exp(x)^j.

a(n) = C(n,2) * Sum_{j=0..2} Stirling2(2,j) * j^(n-2).

A273653 Number of forests of labeled rooted trees of height at most 1, with n labels, three of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

5, 40, 220, 1040, 4550, 19040, 77448, 308640, 1209450, 4668400, 17766892, 66718288, 247397150, 906584000, 3285842960, 11788924992, 41902786770, 147668053200, 516315206260, 1792304871280, 6180666260230, 21184488791840, 72205377800600, 244837696095200

Offset: 3

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A143396.

a:= n-> binomial(n,3)*add(Stirling2(3,j)*j^(n-3), j=0..3):
seq(a(n), n=3..40);

E.g.f.: x^3/3! * Sum_{j=0..3} Stirling2(3,j)*exp(x)^j.

a(n) = C(n,3) * Sum_{j=0..3} Stirling2(3,j) * j^(n-3).

A273654 Number of forests of labeled rooted trees of height at most 1, with n labels, four of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

15, 185, 1485, 9905, 59850, 341082, 1872990, 10032990, 52814025, 274436305, 1411449039, 7195628895, 36390053700, 182622509860, 909578754540, 4496410556268, 22062707386875, 107464431586995, 519700280676885, 2495827315731365, 11905791978208590, 56429414876395550

Offset: 4

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A143396.

a:= n-> binomial(n,4)*add(Stirling2(4,j)*j^(n-4), j=0..4):
seq(a(n), n=4..40);

E.g.f.: x^4/4! * Sum_{j=0..4} Stirling2(4,j)*exp(x)^j.

a(n) = C(n,4) * Sum_{j=0..4} Stirling2(4,j) * j^(n-4).

A273655 Number of forests of labeled rooted trees of height at most 1, with n labels, five of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

52, 906, 9891, 87416, 686826, 5020092, 35006202, 236460312, 1562226237, 10158790642, 65294141913, 415967090448, 2631510433188, 16551428673528, 103578621308388, 645186720537744, 4000912828433199, 24700465510596414, 151810205132056779, 928768921758036744

Offset: 5

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A143396.

a:= n-> binomial(n,5)*add(Stirling2(5,j)*j^(n-5), j=0..5):
seq(a(n), n=5..40);

E.g.f.: x^5/5! * Sum_{j=0..5} Stirling2(5,j)*exp(x)^j.

a(n) = C(n,5) * Sum_{j=0..5} Stirling2(5,j) * j^(n-5).

A273656 Number of forests of labeled rooted trees of height at most 1, with n labels, six of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

203, 4718, 66808, 750120, 7370580, 66562188, 568136184, 4663350120, 37228965774, 291297376370, 2245876801168, 17126594554160, 129527848747752, 973416393927768, 7278979868039760, 54211596845506800, 402386221073227314, 2977842560384800878, 21977111826224423176

Offset: 6

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A143396.

a:= n-> binomial(n,6)*add(Stirling2(6,j)*j^(n-6), j=0..6):
seq(a(n), n=6..40);

E.g.f.: x^6/6! * Sum_{j=0..6} Stirling2(6,j)*exp(x)^j.

a(n) = C(n,6) * Sum_{j=0..6} Stirling2(6,j) * j^(n-6).

A273657 Number of forests of labeled rooted trees of height at most 1, with n labels, seven of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

877, 26104, 463212, 6409320, 76595310, 832521096, 8480449692, 82506827832, 776400941745, 7128606776720, 64261876786536, 571350882111984, 5026991629995036, 43878686837771760, 380672326962043560, 3287071510635218064, 28280022744131688519, 242602886107926614952

Offset: 7

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A143396.

a:= n-> binomial(n,7)*add(Stirling2(7,j)*j^(n-7), j=0..7):
seq(a(n), n=7..40);

E.g.f.: x^7/7! * Sum_{j=0..7} Stirling2(7,j)*exp(x)^j.

a(n) = C(n,7) * Sum_{j=0..7} Stirling2(7,j) * j^(n-7).

A273658 Number of forests of labeled rooted trees of height at most 1, with n labels, eight of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

4140, 153063, 3315645, 55239525, 786082275, 10081916559, 120278261103, 1361959159275, 14838761544750, 157056039953670, 1626124163724918, 16555067600440590, 166368479811851850, 1655233308679110930, 16341232769733507570, 160366537782578273850

Offset: 8

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A143396.

a:= n-> binomial(n,8)*add(Stirling2(8,j)*j^(n-8), j=0..8):
seq(a(n), n=8..40);

E.g.f.: x^8/8! * Sum_{j=0..8} Stirling2(8,j)*exp(x)^j.

a(n) = C(n,8) * Sum_{j=0..8} Stirling2(8,j) * j^(n-8).

A273659 Number of forests of labeled rooted trees of height at most 1, with n labels, nine of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

21147, 948280, 24564100, 483787480, 8061693640, 120120836836, 1653807555400, 21489660580960, 267385444371760, 3218688299529560, 37768351403903120, 434451231603262640, 4920317819371507280, 55047363589643246360, 609969295304994614000, 6708182596868603027936

Offset: 9

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A143396.

a:= n-> binomial(n,9)*add(Stirling2(9,j)*j^(n-9), j=0..9):
seq(a(n), n=9..40);

E.g.f.: x^9/9! * Sum_{j=0..9} Stirling2(9,j)*exp(x)^j.

a(n) = C(n,9) * Sum_{j=0..9} Stirling2(9,j) * j^(n-9).

A273660 Number of forests of labeled rooted trees of height at most 1, with n labels, ten of which are used for root nodes and any root may contain >= 1 labels.

Original entry on oeis.org

115975, 6188545, 188526162, 4324211606, 83234980829, 1422697691415, 22336690031656, 329296289467288, 4628083121855622, 62687799439000450, 824939949711312292, 10611095747493196956, 134039591449028959694, 1668916223557034160170, 20541239452587941670792

Offset: 10

Alois P. Heinz, May 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..986

Column k=10 of A143396.

a:= n-> binomial(n,10)*add(Stirling2(10,j)*j^(n-10), j=0..10):
seq(a(n), n=10..40);

E.g.f.: x^10/10! * Sum_{j=0..10} Stirling2(10,j)*exp(x)^j.

a(n) = C(n,10) * Sum_{j=0..10} Stirling2(10,j) * j^(n-10).

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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A273652 Number of forests of labeled rooted trees of height at most 1, with n labels, two of which are used for root nodes and any root may contain >= 1 labels.

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A273653 Number of forests of labeled rooted trees of height at most 1, with n labels, three of which are used for root nodes and any root may contain >= 1 labels.

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Maple

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A273654 Number of forests of labeled rooted trees of height at most 1, with n labels, four of which are used for root nodes and any root may contain >= 1 labels.

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Maple

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A273655 Number of forests of labeled rooted trees of height at most 1, with n labels, five of which are used for root nodes and any root may contain >= 1 labels.

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Maple

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A273656 Number of forests of labeled rooted trees of height at most 1, with n labels, six of which are used for root nodes and any root may contain >= 1 labels.

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Maple

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A273657 Number of forests of labeled rooted trees of height at most 1, with n labels, seven of which are used for root nodes and any root may contain >= 1 labels.

Views

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Crossrefs

Programs

Maple

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A273658 Number of forests of labeled rooted trees of height at most 1, with n labels, eight of which are used for root nodes and any root may contain >= 1 labels.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A273659 Number of forests of labeled rooted trees of height at most 1, with n labels, nine of which are used for root nodes and any root may contain >= 1 labels.

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