A292889 -id:A292889 - cp's OEIS frontend

A145453 Exponential transform of binomial(n,3) = A000292(n-2).

1, 0, 0, 1, 4, 10, 30, 175, 1176, 7084, 42120, 286605, 2270180, 19213766, 166326524, 1497096055, 14374680880, 147259920760, 1582837679056, 17659771122969, 204674606377140, 2473357218561250, 31148510170120420, 407154732691440811, 5504706823227724904

Offset: 0

Alois P. Heinz, Oct 10 2008

nonn

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.

a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.

Seiichi Manyama, Table of n, a(n) for n = 0..530 (terms 0..200 from Alois P. Heinz)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

3rd column of A145460, A143398.

Cf. A292889, A292910.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);

Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

E.g.f.: exp(exp(x)*x^3/3!).

A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 2, 10, 15, 1, 0, 0, 6, 41, 52, 1, 0, 0, 6, 24, 196, 203, 1, 0, 0, 0, 24, 140, 1057, 877, 1, 0, 0, 0, 24, 60, 870, 6322, 4140, 1, 0, 0, 0, 0, 120, 480, 5922, 41393, 21147, 1, 0, 0, 0, 0, 120, 360, 5250, 45416, 293608, 115975

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   1,  1,  0,  0,  0, ...
   2,  3,  2,  0,  0, ...
   5, 10,  6,  6,  0, ...
  15, 41, 24, 24, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292973.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292978(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292978(20)

T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022

A292908 Expansion of e.g.f. exp(x^3 * exp(-x)).

Original entry on oeis.org

1, 0, 0, 6, -24, 60, 240, -4830, 39984, -180936, -605520, 24616350, -318005160, 2385790836, -86488584, -350543790870, 6917020827360, -79778558317200, 357117438258144, 9237998478286134, -304182278908538040, 5166739059890655660, -48968796671246843160

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..505

Column k=3 of A292973.

Cf. A292889.

x='x+O('x^66); Vec(serlaplace(exp(x^3*exp(-x))))

a(n) = n!*sum(k=0, n\3, (-k)^(n-3*k)/(k!*(n-3*k)!)); \\ Seiichi Manyama, Jul 10 2022

a(n) = n! * Sum_{k=0..floor(n/3)} (-k)^(n-3*k)/(k! * (n-3*k)!). - Seiichi Manyama, Jul 10 2022

A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 840, 10290, 80976, 847224, 13306320, 190271070, 2677088040, 46082426676, 874515884424, 16582066303530, 336875275380000, 7539189088358640, 176554878235711776, 4295134487197296054, 111114287924643309240, 3036073975138066955820

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..431

Cf. A006153, A358080.

Cf. A292889, A355575, A358065.

With[{nn=30},CoefficientList[Series[1/(1-x^3 Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*exp(x))))

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(n-3*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/(n - 3*k)!.

a(n) ~ n! / ((1 + LambertW(1/3)) * 3^(n+1) * LambertW(1/3)^n). - Vaclav Kotesovec, Oct 30 2022

A292891 Expansion of e.g.f. exp(x^3 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 20496, 181944, 1059120, 4990590, 100458600, 1634594676, 18436740504, 164378216730, 2124284725920, 38171412643440, 631390188466656, 8760417873485814, 124649582165430840, 2167585391936047020, 41833303600025220360

Offset: 0

Seiichi Manyama, Sep 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Column k=3 of A292892.

Cf. A292889, A354001.

With[{nn=30},CoefficientList[Series[Exp[x^3 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2022 *)

x='x+O('x^66); Vec(serlaplace(exp(x^3*(exp(x)-1))))

a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(n-3*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, j/(j-3)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/(n-3*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k/(k-3)! * a(n-k)/(n-k)!. (End)

A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 120960, 1874880, 34473600, 738460800, 17982518400, 489858969600, 14834839219200, 498452777222400, 18583796335104000, 768773914900992000, 35220800475250790400, 1779227869201400217600, 98469904378626772992000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..323

Cf. A345747, A354436.

Cf. A292889, A352945.

Join[{1}, Table[n!*Sum[k^(n - 3*k)/k!, {k, 0, n/3}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

PARI
```
a(n) = n!*sum(k=0, n\3, k^(n-3*k)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^(3*k)/(k!*(1-k*x)))))

E.g.f.: Sum_{k>=0} x^(3*k) / (k! * (1 - k * x)).

a(n) ~ sqrt(Pi) * exp((n - 1/2)/LambertW(exp(3/4)*(2*n - 1)/8) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(3/4)*(2*n - 1)/8)) * 2^(2*n + 1/2) * LambertW(exp(3/4)*(2*n - 1)/8)^n). - Vaclav Kotesovec, Oct 30 2022

A362703 Expansion of e.g.f. 1/(1 + LambertW(-x^3 * exp(x))).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 1560, 20370, 161616, 2601144, 53827920, 829605150, 14894289960, 360575394036, 8234733389064, 188800085076330, 5145737430116640, 148419618327231600, 4278452209330445856, 134018446273097264694, 4529883358179857555640

Offset: 0

Seiichi Manyama, Apr 30 2023

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362702.

Cf. A292889, A362348, A362699.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^3*exp(x)))))

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-2*k) / (k! * (n-3*k)!).

A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

Original entry on oeis.org

0, 0, 0, 6, 24, 60, 480, 5250, 40656, 363384, 4839120, 65198430, 859543080, 13311494196, 233478687624, 4190929145130, 79746180437280, 1667320408619760, 36965002127643936, 854734007793179574, 20962277675893792440, 544839141515795731500

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366459.

Cf. A292889, A346754, A358081.

a(n) = n!*sum(k=1, n\3, k^(n-3*k-1)/(n-3*k)!);

a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(n-3*k)!.

cp's OEIS Frontend

A145453 Exponential transform of binomial(n,3) = A000292(n-2).

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A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

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A292908 Expansion of e.g.f. exp(x^3 * exp(-x)).

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A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

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A292891 Expansion of e.g.f. exp(x^3 * (exp(x) - 1)).

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A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

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Mathematica

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A362703 Expansion of e.g.f. 1/(1 + LambertW(-x^3 * exp(x))).

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PARI

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A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

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