A060850 -id:A060850 - cp's OEIS frontend

A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 28, 25, 30, 40, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256

Offset: 0

Alford Arnold, Mar 31 2007

The tree begins (at height n, n >= 0, nodes represent partitions of n)
0:  1
1:  2
2:  3  4
3:  5  6  8
4:  7 10  9 12 16
5: 11 14 15 20 18 24 32
...
and hence differs from A114622.
Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018
These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

			The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
From _Daniel Forgues_, Aug 07 2018: (Start)
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
From _Gus Wiseman_, May 19 2020: (Start)
The sequence together with the corresponding partitions begins:
    1: ()            24: (2,1,1,1)         35: (4,3)
    2: (1)           32: (1,1,1,1,1)       42: (4,2,1)
    3: (2)           13: (6)               56: (4,1,1,1)
    4: (1,1)         22: (5,1)             50: (3,3,1)
    5: (3)           21: (4,2)             45: (3,2,2)
    6: (2,1)         28: (4,1,1)           60: (3,2,1,1)
    8: (1,1,1)       25: (3,3)             80: (3,1,1,1,1)
    7: (4)           30: (3,2,1)           54: (2,2,2,1)
   10: (3,1)         40: (3,1,1,1)         72: (2,2,1,1,1)
    9: (2,2)         27: (2,2,2)           96: (2,1,1,1,1,1)
   12: (2,1,1)       36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
   16: (1,1,1,1)     48: (2,1,1,1,1)       19: (8)
   11: (5)           64: (1,1,1,1,1,1)     34: (7,1)
   14: (4,1)         17: (7)               39: (6,2)
   15: (3,2)         26: (6,1)             52: (6,1,1)
   20: (3,1,1)       33: (5,2)             55: (5,3)
   18: (2,2,1)       44: (5,1,1)           66: (5,2,1)
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).
OEIS Wiki, Partitions#Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order.

Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).
Row lengths are A000041.
Compositions under the same order are A066099.
The opposite version (sum/lex) is A334434.
The length-sensitive version (sum/length/revlex) is A334438.
The version for reversed (weakly increasing) partitions is A334436.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Sum of prime indices is A056239.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.

b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
                b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

From Gus Wiseman, May 19 2020: (Start)
A001222(a(n)) = A238966(n).
A001221(a(n)) = A115623(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A331581(n).
(End)

A128628 An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, 1, 2, 2, 3, 3, 4, 5, 1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 5, 6, 7, 1, 2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 4, 5, 6, 7, 8, 1, 2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 3, 3, 4, 4, 5, 6, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9

Offset: 1

Alford Arnold, Mar 27 2007, Aug 01 2007

The next level gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent.
Sequence A036043 counts the parts of numeric partitions and contains the same values on each row as the current sequence. When a node generates two branches the first branch can be mapped to cyclic partitions; all other branches map to matching partitions.
Appears to be the triangle in which the n-th row contains the number of parts of each partition of n, where the partitions are ordered as in A080577. - Jason Kimberley, May 12 2010

			The values at level three are 1, 2, and 3.
The 1 generates 1 and 2; the 2 generates 2 and 3; the 3 only generates 4.
The array begins
1
1 2
1 2 3
1 2 2 3 4
1 2 2 3 3 4 5
1 2 2 3 2 3 4 3 4 5 6

T. D. Noe, Rows n = 1..25 of irregular triangle, flattened

Cf. A000041, A060850, A128629.
Cf. A006128 (row sums), A036043.
Cf. A177740.
Cf. A308355 (limiting row sequence).

Flatten[Table[Length /@ IntegerPartitions[n], {n, 9}]] (* T. D. Noe, Feb 27 2014 *)

A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds.

Original entry on oeis.org

22, 185, 810, 2580, 6765, 15525, 32305, 62337, 113265, 195910, 325193, 521235, 810654, 1228080, 1817910, 2636326, 3753600, 5256711, 7252300, 9869990, 13266099, 17627775, 23177583, 30178575, 38939875, 49822812, 63247635

Offset: 1

Alford Arnold, Apr 03 2007

A slightly different method of calculating this sequence is described in A128627.

			From A128629 we can construct the table below:
      Deg    #  Associated sequence
  -------  ---  -------------------
        8    1    1  1   2   3    4
       44    2    3  1   3   6   10
       53   11    4  1   4   9   16
       62   11    4  1   4   9   16
       71   11    4  1   4   9   16
      332   12    6  1   6  18   40
      422   12    6  1   6  18   40
      431  111    8  1   8  27   64
      521  111    8  1   8  27   64
      611   12    6  1   6  18   40
     2222    4    7  1   5  15   35
     3221  112   12  1  12  54  160
     3311   22    9  1   9  36  100
     4211  112   12  1  12  54  160
     5111   13   10  1   8  30   80
    22211   23   15  1  12  60  200
    32111  113   20  1  16  90  320
    41111   14   14  1  10  45  140
   221111   24   21  1  15  90  350
   311111   15   22  1  12  63  224
  1111111    8   19  1   9  45  165
  2111111   16   26  1  14  84  336
  -------  ---   -- -- --- --- ----
              Sums: 22 185 810 2580 ...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A000041, A000712, A000716, A023003, A060850, A128627, A128629.

with (numtheory): b:=proc(n) option remember; local d, j; `if` (n=0, 1, add (add (d, d=divisors(j)) *b(n-j), j=1..n)/n) end: A:= proc (n) option remember; local k; `if` (n=0, x, expand (add (b(k-1) *A(n-k) *x^(k-1), k=1..n))) end: a:= n-> coeftayl (A(n+8), x=0, 9): seq(a(n), n=1..40); # Alois P. Heinz, Oct 16 2008
# second Maple program:
a:= n-> n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 17 2008

LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {22, 185, 810, 2580, 6765, 15525, 32305, 62337, 113265}, 30] (* Jean-François Alcover, Mar 07 2021 *)

From Alois P. Heinz, Oct 17 2008: (Start)
G.f.: x*(x-2)*(2*x^5-14*x^4+35*x^3-32*x^2-x+11)/(x-1)^9.
a(n) = n*(n+6)*(n+3)*(n+1)*(4200+(9994+(1571+(74+n)*n)*n)*n)/40320. (End)

More terms from Alois P. Heinz, Oct 16 2008

A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 69, 165, 393, 937, 2233, 5322, 12683, 30227, 72037, 171680, 409151, 975097, 2323870, 5538294, 13198973, 31456058, 74966710, 178662171, 425791279, 1014754341, 2418382956, 5763538903, 13735781840, 32735391558, 78015643589

Offset: 0

Alford Arnold, Feb 05 2002

nonn

Previous name was: Invert transform of right-shifted partition function (A000041).
Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....
Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008
Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009
From L. Edson Jeffery, Mar 16 2011: (Start)
Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)
  (1,  0, ...            )
  (1,  1, 0, ...         )
  (2,  2, 1, 0, ...      )
  (3,  5, 3, 1, 0, ...   )
  (5, 10, 9, 4, 1, 0, ...)
etc., and a(n) is the sum of entries in row n of T. (End)

			The array begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  0,  1,  2,   3,   4,  5,  6, 7, ...
  0,  2,  5,   9,  14, 20, 27, ...
  0,  3, 10,  22,  40, 65, ...
  0,  5, 20,  51, 105, ...
  0,  7, 36, 108, ...
  0, 11, 65, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
N. J. A. Sloane, Transforms

Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.
Cf. table A060850.
Cf. A137682, A143866.
Antidiagonal sums of A144064.

N=66; x='x+O('x^N); et=eta(x); Vec( sum(n=0,N, x^n/et^n ) ) \\ Joerg Arndt, May 08 2009

a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - Vladeta Jovovic, Apr 07 2003
O.g.f.: 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ c / r^n, where r = A347968 = 0.419600352598356478498775753566700025318... is the root of the equation QPochhammer(r) = r and c = 0.3777957165566422058901624844315414446044096308877617181754... = Log[r]/(Log[(1 - r)*r] + QPolyGamma[1, r] - Log[r]*Derivative[0, 1][QPochhammer][r, r]). - Vaclav Kotesovec, Feb 16 2017, updated Mar 31 2018

More terms from Vladeta Jovovic, Apr 07 2003
More terms and better definition from Franklin T. Adams-Watters, Mar 14 2006
New name (using g.f. by Vladimir Kruchinin), Joerg Arndt, Feb 19 2014

A128627 Triangle read by rows. Convolution triangle based on A002865.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 2, 5, 3, 4, 0, 1, 4, 6, 9, 4, 5, 0, 1, 4, 13, 12, 14, 5, 6, 0, 1, 7, 16, 28, 20, 20, 6, 7, 0, 1, 8, 30, 39, 50, 30, 27, 7, 8, 0, 1, 12, 40, 78, 76, 80, 42, 35, 8, 9, 0, 1, 14, 66, 115, 161, 130, 119, 56, 44, 9, 10, 0, 1

Offset: 1

Alford Arnold, Mar 22 2007

Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds (cf. A060850).
The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences).
    1     2     3
   1     3     6
  4'     2     5     9
    1     2     3     4
   1     4     9    16
  5'     2     6    12    20
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     4    10    20    35    56    84   120   165
  6'     4    13    28    50    80   119   168   228   300
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
  1     6    18    40    75   126   196   288   405
  7'     4    16    39    76   130   204   301   424   576
    1     2     3     4     5     6     7     8     9
   1     4     9    16    25    36    49    64    81
   1     4     9    16    25    36    49    64    81
   1     3     6    10    15    21    28    36    45
  1     6    18    40    75   126   196   288   405
  1     6    18    40    75   126   196   288   405
 1     5    15    35    70   126   210   330   495
  8'     7    30    78   161   290   477   735  1078  1521

			The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated:
   1
   0    1
   1    0    1
   1    2    0    1
   2    2    3    0
   2    5    3    4
   4    6    9    4
   4   13   12   14
   7   16   28   20
       30   39   50
            78   76
                161
times
   1
   1    9
   1    8   45
   1    7   36  165
   1    6   28  120
   1    5   21   84
   1    4   15   56
   1    3   10   35
   1    2    6   20
        1    3   10
             1    4
                  1
yields
   1
   0    9
   1    0   45
   1   14    0  165
   2   12   84    0
   2   25   63  336
   4   24  135  224
   4   39  120  490
   7   32  168  400
       30  117  500
            78  304
                161
summing to
  22  185  810 2580 ...
Triangle T(n, k) starts:
  [ 1] 1;
  [ 2] 0,  1;
  [ 3] 1,  0,  1;
  [ 4] 1,  2,  0,  1;
  [ 5] 2,  2,  3,  0,  1;
  [ 6] 2,  5,  3,  4,  0,  1;
  [ 7] 4,  6,  9,  4,  5,  0,  1;
  [ 8] 4, 13, 12, 14,  5,  6,  0,  1;
  [ 9] 7, 16, 28, 20, 20,  6,  7,  0,  1;
  [10] 8, 30, 39, 50, 30, 27,  7,  8,  0,  1;

Cf. A002865, A060850, A007318.

# Using function A002865 and function PMatrix from A357368.
A128627Triangle := proc(dim) local M, Row, r;
M := PMatrix(dim, n -> A002865(n-1));
Row := r -> convert(linalg:-row(M, r), list)[2..r];
for r from 2 to dim do lprint(Row(r)) od end:
A128627Triangle(11); # Peter Luschny, Oct 03 2022

New name by Peter Luschny, Oct 03 2022

cp's OEIS Frontend

A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

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A128628 An irregular triangular array read by rows, with shape sequence A000041(n) related to sequence A060850.

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A129126 Ninth diagonal of table A060850 counting partitions into parts of k kinds.

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A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

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A128627 Triangle read by rows. Convolution triangle based on A002865.

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