A290354 -id:A290354 - cp's OEIS frontend

A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 6, 5, 1, 0, 1, 1, 5, 10, 14, 7, 1, 0, 1, 1, 6, 15, 30, 27, 11, 1, 0, 1, 1, 7, 21, 55, 75, 58, 15, 1, 0, 1, 1, 8, 28, 91, 170, 206, 111, 22, 1, 0, 1, 1, 9, 36, 140, 336, 571, 518, 223, 30, 1, 0

Offset: 0

Alois P. Heinz, Jul 28 2017

A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k.  A(3,3) = 6:
:    o        o       o        o        o       o
:    |        |       |       / \      / \     /|\
:    o        o       o      o   o    o   o   o o o
:    |       / \     /|\     |   |   ( )  |   | | |
:    o      o   o   o o o    o   o   o o  o   o o o
:   /|\    ( )  |   | | |   ( )  |   | |  |   | | |
:  o o o   o o  o   o o o   o o  o   o o  o   o o o

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  0, 1,  2,   3,    4,    5,     6,     7,      8, ...
  0, 1,  3,   6,   10,   15,    21,    28,     36, ...
  0, 1,  5,  14,   30,   55,    91,   140,    204, ...
  0, 1,  7,  27,   75,  170,   336,   602,   1002, ...
  0, 1, 11,  58,  206,  571,  1337,  2772,   5244, ...
  0, 1, 15, 111,  518, 1789,  5026, 12166,  26328, ...
  0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
Index entries for sequences related to rooted trees

Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.
Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
Main diagonal gives A290354.
Cf. A144150.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).

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).

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A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

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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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A305725 a(n) is the n-th term of the sequence that shifts left by one position when Euler transform is applied n times; a(0) = 0.

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