A096651 -id:A096651 - cp's OEIS frontend

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
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]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

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[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

A096752 Number of n-dimensional partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 45, 216, 1197, 7134, 46209, 319555, 2350183, 18254380, 149117618, 1275857233, 11396595255

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal of A096751.

Cf. A096751, A096651.

a(n) = n-th row sums of A096651^n, with a(0)=1.

A179855 Number of 8-dimensional partitions of n.

Original entry on oeis.org

1, 9, 45, 201, 819, 3231, 12321, 46209, 170370, 621316, 2240838, 8011584, 28395213, 99845553, 348333411, 1205925033, 4142850423

Offset: 1

Suresh Govindarajan, Jan 11 2011

nonn

Andrews, George E., The Theory of Partitions, Cambridge University Press, 1984. Cambridge Books Online. Cambridge University Press.

Cf. A000041, A000219, A000293, A000334, A000390, A000416, A000427, A096651.

A096652 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (2n)-dimensional partition numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 5, 2, 1, 0, 7, 7, 7, 2, 1, 0, 11, 16, 9, 9, 2, 1, 0, 15, 15, 31, 11, 11, 2, 1, 0, 22, 59, -4, 54, 13, 13, 2, 1, 0, 30, -109, 313, -72, 87, 15, 15, 2, 1, 0, 42, 1314, -1922, 1122, -225, 132, 17, 17, 2, 1, 0, 56, -11804, 19468, -9671, 3087, -509, 191, 19, 19, 2, 1, 0, 77, 133957, -217176, 110734, -32581

Offset: 0

Paul D. Hanna, Jul 06 2004

Row sums of T form A000219 (planar partitions); row sums of T^2 form A000334(4-D); row sums of T^3 form A000416(6-D).

			Triangle T begins:
{1},
{0,1},
{0,2,1},
{0,3,2,1},
{0,5,5,2,1},
{0,7,7,7,2,1},
{0,11,16,9,9,2,1},
{0,15,15,31,11,11,2,1},
{0,22,59,-4,54,13,13,2,1},
{0,30,-109,313,-72,87,15,15,2,1},
{0,42,1314,-1922,1122,-225,132,17,17,2,1},
{0,56,-11804,19468,-9671,3087,-509,191,19,19,2,1},
{0,77,133957,-217176,110734,-32581,7137,-980,266,21,21,2,1},
{0,101,-1728760,2809257,-1426436,422732,-87714,14601,-1704,359,23,23,2,1},...
Row sums are: {1,1,3,6,13,24,48,86,160,282,500,859,...} (A000219).
T^2 begins:
{1},
{0,1},
{0,4,1},
{0,10,4,1},
{0,26,14,4,1},
{0,59,38,18,4,1},
{0,140,109,50,22,4,1},
{0,307,256,179,62,26,4,1},
{0,684,709,370,273,74,30,4,1},
{0,1464,1240,1683,438,395,86,34,4,1},...
with row sums: {1,1,5,15,45,120,326,835,2145,5345,...} (A000334).

Cf. A096651, A000219, A000334, A000416.

Matrix square of triangle A096651.

A096653 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (3n)-dimensional partition numbers.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 3, 1, 0, 13, 9, 3, 1, 0, 24, 19, 12, 3, 1, 0, 48, 48, 25, 15, 3, 1, 0, 86, 84, 84, 31, 18, 3, 1, 0, 160, 228, 99, 135, 37, 21, 3, 1, 0, 282, 129, 721, 57, 204, 43, 24, 3, 1, 0, 500, 2521, -2267, 2087, -93, 294, 49, 27, 3, 1, 0, 859, -16291, 29876, -13253, 5229, -417, 408, 55, 30, 3, 1, 0, 1479, 199621, -317919

Offset: 0

Paul D. Hanna, Jul 06 2004

Row sums of T form A000293 (solid partitions); row sums of T^2 form A000416(6-D).

			Triangle T begins:
{1},
{0,1},
{0,3,1},
{0,6,3,1},
{0,13,9,3,1},
{0,24,19,12,3,1},
{0,48,48,25,15,3,1},
{0,86,84,84,31,18,3,1},
{0,160,228,99,135,37,21,3,1},
{0,282,129,721,57,204,43,24,3,1},
{0,500,2521,-2267,2087,-93,294,49,27,3,1},
{0,859,-16291,29876,-13253,5229,-417,408,55,30,3,1},
{0,1479,199621,-317919,165456,-46401,11539,-996,549,61,33,3,1},
{0,2485,-2547804,4150781,-2100853,627628,-126896,23006,-1926,720,67,36,3,1},...
Row sums are: {1,1,4,10,26,59,140,307,684,1464,3122,6500,...} (A000293).
T^2 begins:
{1},
{0,1},
{0,6,1},
{0,21,6,1},
{0,71,27,6,1},
{0,216,101,33,6,1},
{0,657,363,131,39,6,1},
{0,1907,1185,552,161,45,6,1},
{0,5507,3931,1824,789,191,51,6,1},
{0,15522,11574,7449,2520,1080,221,57,6,1},...
with row sums: {1,1,7,28,105,357,1197,3857,12300,38430,...} (A000416).

Cf. A096651, A096652, A000293, A000416.

Matrix cube of triangle A096651.

A096799 Triangle, read by rows, where T(n,k) = (k/n)*Sum_{d|n} A096800(d,k).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, -2, 3, 0, 1, 1, 1, -2, 4, 0, 1, 1, -8, 12, -4, 5, 0, 1, 1, 23, -51, 25, -5, 6, 0, 1, 1, -164, 361, -192, 50, -6, 7, 0, 1, 1, 1255, -2856, 1630, -484, 87, -7, 8, 0, 1, 1, -12108, 27795, -16292, 5065, -1026, 140, -8, 9, 0, 1, 1, 136061, -315068, 188665, -60125, 12604, -1925, 212, -9, 10, 0, 1

Offset: 1

Paul D. Hanna, Jul 13 2004

Triangle A096800 lists row polynomials P_n(y), that satisfy the g.f.: A096651(x,y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], with P_n(0)=0 for n>=1; the row sums of A096651^n form the n-dimensional partitions.

			Rows begin:
[1],
[1,1],
[1,0,1],
[1,1,0,1],
[1,-2,3,0,1],
[1,1,-2,4,0,1],
[1,-8,12,-4,5,0,1],
[1,23,-51,25,-5,6,0,1],
[1,-164,361,-192,50,-6,7,0,1],
[1,1255,-2856,1630,-484,87,-7,8,0,1],
[1,-12108,27795,-16292,5065,-1026,140,-8,9,0,1],
[1,136061,-315068,188665,-60125,12604,-1925,212,-9,10,0,1],
[1,-1756686,4093515,-2490800,809665,-173358,27146,-3312,306,-10,11,0,1],...

Cf. A096651, A096800.

A096876 Row sums of the triangle A096875, which transforms n-dimensional partitions into (n-2)-dimensional partitions.

Original entry on oeis.org

1, 1, -1, 0, 1, -1, -1, 3, 4, -17, -27, 118, 267, -917, -3409

Offset: 0

Paul D. Hanna, Jul 13 2004

sign

Cf. A096651, A096874, A096875.

cp's OEIS Frontend

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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A096752 Number of n-dimensional partitions of n.

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A179855 Number of 8-dimensional partitions of n.

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A096652 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (2n)-dimensional partition numbers.

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A096653 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (3n)-dimensional partition numbers.

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A096799 Triangle, read by rows, where T(n,k) = (k/n)*Sum_{d|n} A096800(d,k).

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A096876 Row sums of the triangle A096875, which transforms n-dimensional partitions into (n-2)-dimensional partitions.

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