A096751 -id:A096751 - cp's OEIS frontend

A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 25 2019

An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.

			Array begins:
       k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12
   n=1: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=2: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=3: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=4: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=5: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=6: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=7: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=8: 1    3    6   10   15   21   28   36   45   55   66   78   91
   n=9: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=10: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=11: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=12: 1    4    9   16   25   36   49   64   81  100  121  144  169
  n=13: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=14: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=15: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=16: 1    5   14   30   55   91  140  204  285  385  506  650  819
  n=17: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=18: 1    4    9   16   25   36   49   64   81  100  121  144  169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
  ((2*2*3))          ((2*6))      ((3*4))      ((12))
  ((2)*(2*3))        ((2)*(6))    ((3)*(4))
  ((3)*(2*2))        ((2))*((6))  ((3))*((4))
  ((2))*((2*3))
  ((2)*(2)*(3))
  ((3))*((2*2))
  ((2))*((2)*(3))
  ((3))*((2)*(2))
  ((2))*((2))*((3))

Columns: A001055 (k=1), A050336 (k=2), A050338 (k=3), A050340 (k=4).
Rows: A000027, A000217, A000290, etc.
Cf. A096751, A141268, A144150, A213427, A255906, A281113, A290353, A292504, A317145, A318564, A318565, A318812, A323718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lev[n_,k_]:=If[k==0,{n},Join@@Table[Union[Sort/@Tuples[lev[#,k-1]&/@fac]],{fac,facs[n]}]];
Table[Length[lev[sum-k,k]],{sum,12},{k,0,sum-1}]

A096807 Row sums of triangle A096806, in which the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

Original entry on oeis.org

1, 2, 4, 10, 26, 78, 246, 844, 3062, 11782, 47664, 202254, 896462, 4139514

Offset: 1

Paul D. Hanna, Jul 19 2004

nonn

Cf. A096751, A096806.

A116673 Row sums of triangle A116672, in which the binomial transform of the n-th row lists the Euler transform of the n-th sequence in A007318 (Pascal's Triangle).

Original entry on oeis.org

1, 2, 4, 10, 26, 80, 262, 950

Offset: 1

Alford Arnold, Feb 22 2006

nonn

A116673 is to A096807 as Table A116672 is to Table A096806. The difference between the two tables is of historical interest. (cf. A096751 and A007326).

			A116672 begins
1; 1,1; 1,2,1; 1,4,4,1; 1,6,11,7,1; 1,10,27,29,12,1; 1,14,57,96,72,21,1; 1,21,117,277,319,176,38,1; . . . so
A116673 begins 1 2 4 10 26 80 262 950 ...

Cf. A007318, A007326, A096751, A096806, A096807, A116672.

A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 141, 86, 22, 1, 1, 1, 9, 36, 105, 216, 331, 310, 160, 30, 1, 1, 1, 10, 45, 148, 357, 672, 855, 692, 282, 42, 1, 1, 1, 11, 55, 201, 554, 1232, 1982, 2214, 1483, 500, 56, 1

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.

			Square array begins:
1,  1,   1,   1,    1,    1,  ...
1,  1,   1,   1,    1,    1,  ...
1,  2,   3,   4,    5,    6,  ...
1,  3,   6,  10,   15,   21,  ...
1,  5,  13,  26,   45,   71,  ...
1,  7,  24,  59,  120,  216,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
N. J. A. Sloane, Transforms

Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.
Main diagonal gives A293554.
Cf. A007318, A096751 (a similar but different sequence).

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 17 2017

Table[Function[k, SeriesCoefficient[E^(Sum[x^i/(i (1 - x^i)^k), {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

For asymptotics of column k see comment from Vaclav Kotesovec in A255965.

A096802 Row sums of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

Original entry on oeis.org

1, 1, 2, 4, 12, 42, 188, 950, 5362, 33361, 222120, 1600145, 12069976, 98105654

Offset: 0

Paul D. Hanna, Jul 13 2004

nonn

Cf. A096801, A096751, A096752.

A096803 Column 1 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

Original entry on oeis.org

1, 1, 2, 7, 26, 124, 640, 3695, 23231, 156572, 1133838, 8635777, 70212042

Offset: 0

Paul D. Hanna, Jul 13 2004

nonn

Cf. A096801, A096751, A096752.

A096805 Column 3 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

Original entry on oeis.org

1, 1, 4, 13, 68, 332, 2100, 12566, 94878, 623351, 5828851

Offset: 0

Paul D. Hanna, Jul 13 2004

nonn

Cf. A096801, A096751, A096752.

A096804 Column 2 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

Original entry on oeis.org

1, 1, 3, 10, 44, 218, 1208, 7403, 48663, 346636, 2590866, 20875236

Offset: 0

Paul D. Hanna, Jul 13 2004

nonn

Cf. A096801, A096751, A096752.

cp's OEIS Frontend

A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

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A096807 Row sums of triangle A096806, in which the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

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A116673 Row sums of triangle A116672, in which the binomial transform of the n-th row lists the Euler transform of the n-th sequence in A007318 (Pascal's Triangle).

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A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

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A096802 Row sums of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

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A096803 Column 1 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

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A096805 Column 3 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

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A096804 Column 2 of triangle A096801, which transforms the (n+m)-dimensional partitions of n into the (n+m+1)-dimensional partitions of n, for any fixed m.

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