A261280 -id:A261280 - cp's OEIS frontend

A144150 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where the e.g.f. of column k is 1+g^(k+1)(x) with g = x-> exp(x)-1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 15, 1, 1, 1, 5, 22, 60, 52, 1, 1, 1, 6, 35, 154, 358, 203, 1, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1, 1, 1, 9, 92, 910, 7556, 44590, 146115, 167894, 21147, 1, 1, 1, 10, 117

Offset: 0

Alois P. Heinz, Sep 11 2008

A(n,k) is also the number of (k+1)-level labeled rooted trees with n leaves.
Number of ways to start with set {1,2,...,n} and then repeat k times: partition each set into subsets. - Alois P. Heinz, Aug 14 2015
Equivalently, A(n,k) is the number of length k+1 multichains from bottom to top in the set partition lattice of an n-set. - Geoffrey Critzer, Dec 05 2020

			Square array begins:
  1,  1,   1,    1,    1,    1,  ...
  1,  1,   1,    1,    1,    1,  ...
  1,  2,   3,    4,    5,    6,  ...
  1,  5,  12,   22,   35,   51,  ...
  1, 15,  60,  154,  315,  561,  ...
  1, 52, 358, 1304, 3455, 7556,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
E. T. Bell, The Iterated Exponential Integers, Annals of Mathematics, 39(3) (1938), 539-557.
Pierpaolo Natalini and Paolo Emilio Ricci, Higher order Bell polynomials and the relevant integer sequences, in Appl. Anal. Discrete Math. 11 (2017), 327-339.
Pierpaolo Natalini and Paolo E. Ricci, Integer Sequences Connected with Extensions of the Bell Polynomials, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.
Ivar Henning Skau and Kai Forsberg Kristensen, An asymptotic Formula for the iterated exponential Bell Numbers, arXiv:1903.07979 [math.CO], 2019.
Ivar Henning Skau and Kai Forsberg Kristensen, Sets of iterated Partitions and the Bell iterated Exponential Integers, arXiv:1903.08379 [math.CO], 2019.
Index entries for sequences related to rooted trees

Columns k=0-10 give: A000012, A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A081697, A081740.
Rows n=0+1, 2-5 give: A000012, A000027, A000326, A005945, A005946.
First lower diagonal gives A139383.
First upper diagonal gives A346802.
Main diagonal gives A261280.
Cf. A000142, A111672, A290353.

g:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1
      else (n-1)! *add(p(k)*b(n-k)/(k-1)!/(n-k)!, k=1..n) fi
    end end:
A:= (n,k)-> (g@@k)(1)(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 14 2015
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, 1, `if`(n=0,
      b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
    end:
A:= (n, k)-> b(n, k, 0):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 04 2021

g[k_] := g[k] = Nest[Function[x, E^x - 1], x, k]; a[n_, k_] := SeriesCoefficient[1 + g[k + 1], {x, 0, n}]*n!; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def A(n, k): return 1 if n==0 or k==0 else sum([binomial(n - 1, j - 1)*A(j, k - 1)*A(n - j, k) for j in range(1, n + 1)])
for n in range(51): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 07 2017

E.g.f. of column k: 1 + g^(k+1)(x) with g = x-> exp(x)-1.

Column k+1 is Stirling transform of column k.

A139383 Number of n-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 2, 12, 154, 3455, 120196, 5995892, 406005804, 35839643175, 3998289746065, 550054365477936, 91478394767427823, 18091315306315315610, 4196205472500769304318, 1128136777063831105273242, 347994813261017613045578964, 122080313159891715442898099217

Offset: 0

Paul D. Hanna, Apr 16 2008

nonn

Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. - Gerald McGarvey, Aug 19 2009

			If we form a table from the family of sequences defined by:
number of k-level labeled rooted trees with n leaves,
then this sequence equals the diagonal in that table:
n=1:A000012=[1,1,1,1,1,1,1,1,1,1,...];
n=2:A000110=[1,2,5,15,52,203,877,4140,21147,115975,...];
n=3:A000258=[1,3,12,60,358,2471,19302,167894,1606137,...];
n=4:A000307=[1,4,22,154,1304,12915,146115,1855570,26097835,...];
n=5:A000357=[1,5,35,315,3455,44590,660665,11035095,204904830,...];
n=6:A000405=[1,6,51,561,7556,120196,2201856,45592666,1051951026,...];
n=7:A001669=[1,7,70,910,14532,274778,5995892,148154860,4085619622,...];
n=8:A081624=[1,8,92,1380,25488,558426,14140722,406005804,13024655442,...];
n=9:A081629=[1,9,117,1989,41709,1038975,29947185,979687005,35839643175,..].
Row n in the above table equals column 0 of matrix power A008277^n where A008277 = triangle of Stirling numbers of 2nd kind:
1;
1,1;
1,3,1;
1,7,6,1;
1,15,25,10,1;
1,31,90,65,15,1; ...
The name of this sequence is a generalization of the definition given in the above sequences by _Christian G. Bower_.

Alois P. Heinz, Table of n, a(n) for n = 0..247

Cf. A008277; A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A261280, A290354.

A diagonal of A144150.

A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
a:= n-> A(n, n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2015
# second Maple program:
g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1+(g@@n)(x), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 31 2017
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, `if`(n<2, 1, 0),
     `if`(n=0, b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

t[n_,m_]:=t[n,m] = If[m==1,1,Sum[StirlingS2[n,k]*t[k,m-1],{k,1,n}]]; Table[t[n,n],{n,1,20}] (* Vaclav Kotesovec, Aug 14 2015 after Vladimir Kruchinin *)

T(n,m):=if m=1 then 1 else sum(stirling2(n,i)*T(i,m-1),i,1,n);
makelist(T(n,n),n,1,7); /* Vladimir Kruchinin, May 19 2012 */

{a(n)=local(E=exp(x+x*O(x^n))-1,F=x); for(i=1,n,F=subst(F,x,E));n!*polcoeff(F,n)}

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def A(n, k): return 1 if n==0 or k==0 else sum(binomial(n - 1, j - 1)*A(j, k - 1)*A(n - j, k) for j in range(1, n + 1))
def a(n): return A(n, n - 1)
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 07 2017, after Maple code

a(n) = T(n,n), T(n,m) = Sum_{i=1..n} Stirling2(n,i)*T(i,m-1), m>1, T(n,1)=1. - Vladimir Kruchinin, May 19 2012
a(n) = n! * [x^n] 1 + g^n(x), where g(x) = exp(x)-1. - Alois P. Heinz, Aug 14 2015
From Vaclav Kotesovec, Aug 14 2015: (Start)
Conjecture: a(n) ~ c * n^(2*n-5/6) / (2^(n-1) * exp(n)), where c = 2.86539...
a(n) ~ exp(-1) * A261280(n).
(End)

a(0)=1 prepended by Alois P. Heinz, Jul 31 2017

A306187 Number of n-times partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 65, 371, 3780, 33552, 472971, 5736082, 97047819, 1547576394, 32992294296, 626527881617, 15202246707840, 352290010708120, 9970739854456849, 262225912049078193, 8309425491887714632, 250946978120046026219, 8898019305511325083149

Offset: 0

Alois P. Heinz, Jan 27 2019

nonn

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - Gus Wiseman, Jan 27 2019

			From _Gus Wiseman_, Jan 27 2019: (Start)
The a(1) = 1 through a(3) = 10 partitions:
  (1)  ((2))     (((3)))
       ((11))    (((21)))
       ((1)(1))  (((111)))
                 (((2)(1)))
                 (((11)(1)))
                 (((2))((1)))
                 (((1)(1)(1)))
                 (((11))((1)))
                 (((1)(1))((1)))
                 (((1))((1))((1)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..410
Wikipedia, Partition (number theory)

Main diagonal of A323718.
Cf. A000041, A306188.
Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

ptnlevct[n_,k_]:=Switch[k,0,1,1,PartitionsP[n],_,SeriesCoefficient[Product[1/(1-ptnlevct[m,k-1]*x^m),{m,n}],{x,0,n}]];
Table[ptnlevct[n,n],{n,0,8}] (* Gus Wiseman, Jan 27 2019 *)

a(n) = A323718(n,n).

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Seiichi Manyama, Table of n, a(n) for n = 1..246
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022

Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022

A306188 Number of n-times strict partitions of n.

Original entry on oeis.org

1, 1, 1, 4, 11, 41, 154, 904, 4927, 35398, 234454, 1965976, 16589885, 157974740, 1480736877, 16406078770, 177232251249, 2151696598160, 25726133391191, 346746928400037, 4607758596471426, 67562340652906942, 969200312705046531, 15386051753617360150

Offset: 0

Alois P. Heinz, Jan 27 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..480

Cf. A000009, A261280, A306187.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..25);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0 || k == 0, 1, b[n, i - 1, k] + b[i, i, k - 1] b[n - i, Min[n - i, i - 1], k]]];
a[n_] := b[n, n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

A346802 Number of ways to start with set {1,2,...,n} and then repeat (n+1) times: partition each set into subsets.

Original entry on oeis.org

1, 1, 4, 35, 561, 14532, 558426, 29947185, 2141867440, 197304236151, 22773405820375, 3221070321954212, 548135428211610344, 110514990079832223628, 26057791266228066121614, 7105134240266115177248187, 2218719629100693497237788887, 786736247267010426995743418575

Offset: 0

Alois P. Heinz, Aug 04 2021

nonn

Also the number of (n+2)-level labeled rooted trees with n leaves.

Alois P. Heinz, Table of n, a(n) for n = 0..247

First upper diagonal of A144150.

Cf. A139383, A261280.

a:= n-> (g-> coeff(series(1+(g@@(n+2))(x), x, n+1), x, n)*n!)(x-> exp(x)-1):
seq(a(n), n=0..20);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
a:= n-> A(n, n+1):
seq(a(n), n=0..20);
# third Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=0, 1,
      b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);

b[n_, t_, m_] := b[n, t, m] = If[n == 0, If[t == 0, 1, b[m, t - 1, 0]], m*b[n - 1, t, m] + b[n - 1, t, m + 1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 18 2023, after 3rd Maple program *)

a(n) = n! * [x^n] 1 + g^(n+2)(x), where g(x) = exp(x)-1.
a(n) = A144150(n,n+1).
Conjecture: a(n) ~ c * n^(2*n - 5/6) / (exp(n) * 2^n), where c = 42.345... - Vaclav Kotesovec, Aug 11 2021

A363010 a(n) = n! * [x^n] 1/(1 - f^n(x)), where f(x) = exp(x) - 1.

Original entry on oeis.org

1, 1, 4, 36, 594, 15775, 618838, 33757864, 2448904188, 228290728635, 26617527649365, 3797508644987398, 651082351708066303, 132130157056046918808, 31333332827346731906130, 8587011712002719806274022, 2693586800519167315881703732, 958983405298849163873718493941

Offset: 0

Seiichi Manyama, May 12 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..246

Main diagonal of A363007.
Main diagonal of A153278 (for n>=1).
Cf. A139383, A261280, A346802, A351433.

b:= proc(n, t, m) option remember; `if`(n=0, `if`(t<2, m!,
      b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 12 2023

a(n) = T(n,n), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = n!.

cp's OEIS Frontend

A144150 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where the e.g.f. of column k is 1+g^(k+1)(x) with g = x-> exp(x)-1.

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A139383 Number of n-level labeled rooted trees with n leaves.

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A306187 Number of n-times partitions of n.

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A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

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Programs

Maple

Mathematica

PARI

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A306188 Number of n-times strict partitions of n.

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Keywords

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Programs

Maple

Mathematica

A346802 Number of ways to start with set {1,2,...,n} and then repeat (n+1) times: partition each set into subsets.

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Maple

Mathematica

Formula

A363010 a(n) = n! * [x^n] 1/(1 - f^n(x)), where f(x) = exp(x) - 1.

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