A007714 -id:A007714 - 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).

A000334 Number of 4-dimensional partitions of n.

Original entry on oeis.org

1, 5, 15, 45, 120, 326, 835, 2145, 5345, 13220, 32068, 76965, 181975, 425490, 982615, 2245444, 5077090, 11371250, 25235790, 55536870, 121250185, 262769080, 565502405, 1209096875, 2569270050, 5427963902, 11404408525, 23836421895, 49573316740, 102610460240

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Jan 23 2019: (Start)
The a(1) = 1 through a(3) = 15 four-dimensional partitions, represented as chains of chains of chains of integer partitions:
  (((1)))  (((2)))         (((3)))
           (((11)))        (((21)))
           (((1)(1)))      (((111)))
           (((1))((1)))    (((2)(1)))
           (((1)))(((1)))  (((11)(1)))
                           (((2))((1)))
                           (((1)(1)(1)))
                           (((11))((1)))
                           (((2)))(((1)))
                           (((1)(1))((1)))
                           (((11)))(((1)))
                           (((1))((1))((1)))
                           (((1)(1)))(((1)))
                           (((1))((1)))(((1)))
                           (((1)))(((1)))(((1)))
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Suresh Govindarajan, Table of n, a(n) for n = 1..40
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.

Cf. A000219 (2-dim), A000293 (3-dim), A000390 (5-dim), A096751 (k-dim).

Cf. A002974, A007714, A050340.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[Length[levptns[n,4]],{n,8}] (* Gus Wiseman, Jan 24 2019 *)

More terms from Sean A. Irvine, Nov 14 2010

A050340 Number of ways of factoring n with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 1, 5, 1, 15, 5, 5, 1, 25, 1, 5, 5, 55, 1, 25, 1, 25, 5, 5, 1, 105, 5, 5, 15, 25, 1, 35, 1, 170, 5, 5, 5, 145, 1, 5, 5, 105, 1, 35, 1, 25, 25, 5, 1, 425, 5, 25, 5, 25, 1, 105, 5, 105, 5, 5, 1, 205, 1, 5, 25, 571, 5, 35, 1, 25, 5, 35, 1, 660, 1, 5, 25, 25, 5, 35, 1, 425, 55, 5

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on the prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			4 = (((4))) = (((2*2))) = (((2)*(2))) = (((2))*((2))) = (((2)))*(((2))).

R. J. Mathar, Table of n, a(n) for n = 1..1295

Cf. A001055, A050336-A050341. a(p^k) = A007714. a(A002110) = A000357.

Dirichlet g.f.: Product{n=2..infinity} (1/(1-1/n^s)^A050338(n)).

a(n) = A050341(A101296(n)). - R. J. Mathar, May 26 2017

A055886 Euler transform applied three times to partition triangle A008284.

Original entry on oeis.org

1, 1, 4, 1, 4, 10, 1, 8, 16, 30, 1, 8, 32, 54, 75, 1, 12, 48, 128, 176, 206, 1, 12, 70, 210, 443, 535, 518, 1, 16, 92, 362, 842, 1485, 1585, 1344, 1, 16, 124, 516, 1544, 3075, 4676, 4527, 3357, 1, 20, 152, 770, 2500, 6133, 10622, 14336, 12664, 8429, 1, 20, 190, 1030, 3952, 10718, 22524, 34918, 42426, 34631, 20759

Offset: 1

Christian G. Bower, Jun 09 2000

			  1;
  1, 4;
  1, 4, 10;
  1, 8, 16, 30;
  1, 8, 32, 54, 75;
  ...

Alois P. Heinz, Rows n = 1..150, flattened
N. J. A. Sloane, Transforms

Row sums give A007714.
Main diagonal gives A007713.
Cf. A055884, A055885.

A290355 The sixth Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 6, 21, 91, 336, 1337, 5026, 19193, 71769, 268272, 992676, 3659116, 13400426, 48863017, 177299790, 640713627, 2305930966, 8268556438, 29544196129, 105215495691, 373523546056, 1322096328899, 4666327388034, 16425341129078, 57667752483279, 201967215942032

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Also the number of 6-level rooted trees with n leaves. All n leaves are in level 6. a(2) = 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 o    o o
:   ( )    | |    | |    | |    | |    | |
:   o o    o o    o o    o o    o o    o o

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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

Column k=6 of A290353.

Cf. A007714.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n, 6):
seq(a(n), n=0..30);

b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, 6], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f.: Product_{j>0} 1/(1-x^j)^A007714(j).

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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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A000334 Number of 4-dimensional partitions of n.

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A050340 Number of ways of factoring n with 3 levels of parentheses.

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A055886 Euler transform applied three times to partition triangle A008284.

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A290355 The sixth Euler transform of the sequence with g.f. 1+x.

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