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%I #12 Mar 13 2015 19:13:05
%S 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 21,117,269,282,131,22,1,1,29,223,707,1062,766,244,29,1,1,41,417,1747,
%U 3565,3681,1871,421,37,1,1,55,748,4090,10999,15489,11400,4152,683,46,1,1,76
%N 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 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 S. Govindarajan, <a href="http://arxiv.org/abs/1203.4419">Notes on higher-dimensional partitions</a>, arXiv:1203.4419
%F 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 The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
%e BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
%e Rows begin:
%e [1],
%e [1,1],
%e [1,2,1],
%e [1,4,4,1],
%e [1,6,11,7,1],
%e [1,10,27,28,11,1],
%e [1,14,57,93,64,16,1],
%e [1,21,117,269,282,131,22,1],
%e [1,29,223,707,1062,766,244,29,1],
%e [1,41,417,1747,3565,3681,1871,421,37,1],
%e [1,55,748,4090,10999,15489,11400,4152,683,46,1],
%e [1,76,1326,9219,31828,58975,59433,31802,8483,1054,56,1],
%e [1,100,2284,20095,87490,207735,276230,204072,80664,16162,1561,67,1],
%e [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],...
%e The inverse binomial transform of the diagonals of this triangle begin:
%e [1],
%e [1,1,1],
%e [1,3,4,6,3],
%e [1,5,16,29,49,45,15],
%e [1,9,38,127,289,540,660,420,105],
%e [1,13,90,397,1384,3633,7506,10920,9765,4725,945],
%e [1,20,182,1140,5266,19324,55645,125447,?,?,?,62370,10395],...
%Y Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)
%Y Cf. A119271.
%K nonn,tabl
%O 1,5
%A _Paul D. Hanna_, Jul 19 2004
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