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

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