A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 28, 11, 1, 1, 14, 57, 93, 64, 16, 1, 1, 21, 117, 269, 282, 131, 22, 1, 1, 29, 223, 707, 1062, 766, 244, 29, 1, 1, 41, 417, 1747, 3565, 3681, 1871, 421, 37, 1, 1, 55, 748, 4090, 10999, 15489, 11400, 4152, 683, 46, 1, 1, 76

Offset

1,5

Comments

The n-th row equals the inverse binomial transform of n-th column of square array A096751, for n>=1. The zero-dimensional partition of n is taken to be 1 for all n.

Links

Table of n, a(n) for n=1..68.

S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.

Formula

T(n, 0)=T(n, n-1)=1, T(n, 1)=A000041(n)-1, T(n, n-2)=(n-1)*(n-2)/2+1, for n>=1.

Example

The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
Rows begin:
  [1],
  [1,  1],
  [1,  2,   1],
  [1,  4,   4,    1],
  [1,  6,  11,    7,     1],
  [1, 10,  27,   28,    11,     1],
  [1, 14,  57,   93,    64,    16,      1],
  [1, 21, 117,  269,   282,   131,     22,      1],
  [1, 29, 223,  707,  1062,   766,    244,     29,     1],
  [1, 41, 417, 1747,  3565,  3681,   1871,    421,    37,     1],
  [1, 55, 748, 4090, 10999, 15489,  11400,   4152,   683,    46,    1],
  [1, 76,1326, 9219, 31828, 58975,  59433,  31802,  8483,  1054,   56,   1],
  [1,100,2284,20095, 87490,207735, 276230, 204072, 80664, 16162, 1561,  67, 1],
  [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],
  ...
The inverse binomial transform of the diagonals of this triangle begin:
  [1],
  [1, 1,  1],
  [1, 3,  4,   6,  3],
  [1, 5, 16,  29,  49,   45,   15],
  [1, 9, 38, 127, 289,  540,  660,   420, 105],
  [1,13, 90, 397,1384, 3633, 7506, 10920,9765,4725,945],
  [1,20,182,1140,5266,19324,55645,125447,  ? ,  ? , ?  ,62370,10395],
  ...

Crossrefs

Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)

Cf. A119271.

Sequence in context: A172991 A203906 A274310 * A116672 A161126 A128562

Adjacent sequences: A096803 A096804 A096805 * A096807 A096808 A096809

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 19 2004

Status

approved