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%I #20 Jan 09 2023 02:40:02
%S 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,
%T 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,
%U 36,105,216,326,307,160,30,1,1,1,10,45,148,357,657,835,684,282,42,1
%N Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.
%C 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.
%D G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.
%H Pontus von Brömssen, <a href="/A096751/b096751.txt">Table of n, a(n) for n = 0..275</a> (first 23 antidiagonals)
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
%F T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.
%F Inverse binomial transforms of the columns is given by triangle A096806.
%e n-th row lists n-dimensional partitions; table begins with n=0:
%e [1,1,1,1,1,1,1,1,1,1,1,1,...],
%e [1,1,2,3,5,7,11,15,22,30,42,56,...],
%e [1,1,3,6,13,24,48,86,160,282,500,859,...],
%e [1,1,4,10,26,59,140,307,684,1464,3122,...],
%e [1,1,5,15,45,120,326,835,2145,5345,...],
%e [1,1,6,21,71,216,657,1907,5507,15522,...],
%e [1,1,7,28,105,357,1197,3857,12300,38430,...],
%e [1,1,8,36,148,554,2024,7134,24796,84625,...],
%e [1,1,9,45,201,819,3231,12321,46209,170370,...],
%e [1,1,10,55,265,1165,4927,20155,80920,...],...
%e Array begins:
%e k=0: k=1: k=2: k=3: k=4: k=5: k=6: k=7: k=8:
%e n=0: 1 1 1 1 1 1 1 1 1
%e n=1: 1 1 2 3 5 7 11 15 22
%e n=2: 1 1 3 6 13 24 48 86 160
%e n=3: 1 1 4 10 26 59 140 307 684
%e n=4: 1 1 5 15 45 120 326 835 2145
%e n=5: 1 1 6 21 71 216 657 1907 5507
%e n=6: 1 1 7 28 105 357 1197 3857 12300
%e n=7: 1 1 8 36 148 554 2024 7134 24796
%e n=8: 1 1 9 45 201 819 3231 12321 46209
%e n=9: 1 1 10 55 265 1165 4927 20155 80920
%t trans[x_]:=If[x=={},{},Transpose[x]];
%t 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]}]];
%t Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* _Gus Wiseman_, Jan 27 2019 *)
%Y 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).
%Y Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
%Y Cf. A096651, A096752, A096753.
%Y Cf. A096806.
%Y Cf. A042984.
%Y Cf. A144150, A213427, A290353, A323718, A323719.
%K nonn,tabl
%O 0,9
%A _Paul D. Hanna_, Jul 07 2004
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