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A382818 Square array A(n,k), n > 0, k > 0, read by downward antidiagonals: A(n,k) is the number of columns in all k-compositions of n.

%I A382818 #14 Apr 06 2025 08:45:19
%S A382818 1,2,3,3,11,8,4,24,52,20,5,42,163,227,48,6,65,372,1017,944,112,7,93,
%T A382818 710,3019,6030,3800,256,8,126,1208,7095,23256,34563,14944,576,9,164,
%U A382818 1897,14340,67251,173076,193392,57748,1280,10,207,2808,26082,161394,615630,1256936,1062756,220128,2816
%N A382818 Square array A(n,k), n > 0, k > 0, read by downward antidiagonals: A(n,k) is the number of columns in all k-compositions of n.
%C A382818 A k-composition of n is a rectangular array of nonnegative integers with k rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.
%H A382818 John Tyler Rascoe, <a href="/A382818/b382818.txt">Antidiagonals n = 1..100, flattened</a>
%F A382818 Column k has g.f.: -((1 - x)^k - 1)*(1 - x)^k/(((1 - x)^k - 1) + (1 - x)^k)^2.
%e A382818 Square array begins:
%e A382818    1,   2,    3,     4,     5,      6, ...
%e A382818    3,  11,   24,    42,    65,     93, ...
%e A382818    8,  52,  163,   372,   710,   1208, ...
%e A382818   20, 227, 1017,  3019,  7095,  14340, ...
%e A382818   48, 944, 6030, 23256, 67251, 161394, ...
%e A382818   ...
%e A382818 A(2,2) = 11 counts the columns in the 2-compositions of 2:
%e A382818  [2]   [0]   [1]   [1,0]   [0,1]   [0,0]   [1,1]
%e A382818  [0],  [2],  [1],  [0,1],  [1,0],  [1,1],  [0,0].
%o A382818 (PARI)
%o A382818 A382818_Column(k,N) = {my(x='x+O('x^N)); Vec(-(((1 - x)^k - 1)*(1 - x)^k)/( ((1 - x)^k - 1) + (1 - x)^k)^2)}
%o A382818 A382818_array(max_row) = {my(m=matrix(0)); for(n=1,max_row, m=matconcat([m,A382818_Column(n,max_row)~])); m}
%o A382818 A382818_array(10)
%Y A382818 C.f. A001792 (column k=1), A005475 (row n=2), A145839, A181289, A181290 (column k=2), A382820 (main diagonal).
%K A382818 nonn,easy,tabl
%O A382818 1,2
%A A382818 _John Tyler Rascoe_, Apr 05 2025