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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.

%I A096806 #19 Jan 22 2025 06:02:51
%S A096806 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,
%T A096806 21,117,269,282,131,22,1,1,29,223,707,1062,766,244,29,1,1,41,417,1747,
%U A096806 3565,3681,1871,421,37,1,1,55,748,4090,10999,15489,11400,4152,683,46,1,1,76
%N 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.
%C A096806 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.
%H A096806 S. Govindarajan, <a href="http://arxiv.org/abs/1203.4419">Notes on higher-dimensional partitions</a>, arXiv:1203.4419 [math.CO], 2012.
%F A096806 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.
%e A096806 The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
%e A096806 BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
%e A096806 Rows begin:
%e A096806   [1],
%e A096806   [1,  1],
%e A096806   [1,  2,   1],
%e A096806   [1,  4,   4,    1],
%e A096806   [1,  6,  11,    7,     1],
%e A096806   [1, 10,  27,   28,    11,     1],
%e A096806   [1, 14,  57,   93,    64,    16,      1],
%e A096806   [1, 21, 117,  269,   282,   131,     22,      1],
%e A096806   [1, 29, 223,  707,  1062,   766,    244,     29,     1],
%e A096806   [1, 41, 417, 1747,  3565,  3681,   1871,    421,    37,     1],
%e A096806   [1, 55, 748, 4090, 10999, 15489,  11400,   4152,   683,    46,    1],
%e A096806   [1, 76,1326, 9219, 31828, 58975,  59433,  31802,  8483,  1054,   56,   1],
%e A096806   [1,100,2284,20095, 87490,207735, 276230, 204072, 80664, 16162, 1561,  67, 1],
%e A096806   [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],
%e A096806   ...
%e A096806 The inverse binomial transform of the diagonals of this triangle begin:
%e A096806   [1],
%e A096806   [1, 1,  1],
%e A096806   [1, 3,  4,   6,  3],
%e A096806   [1, 5, 16,  29,  49,   45,   15],
%e A096806   [1, 9, 38, 127, 289,  540,  660,   420, 105],
%e A096806   [1,13, 90, 397,1384, 3633, 7506, 10920,9765,4725,945],
%e A096806   [1,20,182,1140,5266,19324,55645,125447,  ? ,  ? , ?  ,62370,10395],
%e A096806   ...
%Y A096806 Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)
%Y A096806 Cf. A119271.
%K A096806 nonn,tabl
%O A096806 1,5
%A A096806 _Paul D. Hanna_, Jul 19 2004