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A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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