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A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.

%I A382923 #15 Apr 14 2025 07:39:26
%S A382923 1,0,1,0,2,3,0,3,5,7,0,4,13,16,16,0,5,14,33,40,35,0,6,29,70,105,100,
%T A382923 75,0,7,27,88,207,292,244,159,0,8,51,152,336,604,758,576,334,0,9,44,
%U A382923 206,588,1161,1749,1920,1329,696,0,10,79,300,882,2076,3685,4924,4802,3028,1442
%N A382923 Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.
%C A382923 For some m > 0, an m-composition of n is a rectangular array of nonnegative integers with m rows, at least one nonzero entry in each column, and having the sum of all entries equal to n.
%H A382923 John Tyler Rascoe, <a href="/A382923/b382923.txt">Antidiagonals n = 0..30, flattened</a>
%F A382923 G.f.: G(t,x) = 1 + Sum_{m>0} -1 + 1/(1 + t^m - (t + x/(1 - x))^m).
%e A382923 Square array begins:
%e A382923    1,   0,   0,   0,    0,    0, ...
%e A382923    1,   2,   3,   4,    5,    6, ...
%e A382923    3,   5,  13,  14,   29,   27, ...
%e A382923    7,  16,  33,  70,   88,  152, ...
%e A382923   16,  40, 105, 207,  336,  588, ...
%e A382923   35, 100, 292, 604, 1161, 2076, ...
%e A382923   ...
%e A382923 A(2,0) = 3 counts:
%e A382923   [2],  [1,1],  [1]
%e A382923                 [1].
%e A382923 A(2,1) = 5 counts:
%e A382923   [2]   [0]   [1]   [1]   [0]
%e A382923   [0],  [2],  [1]   [0]   [1]
%e A382923               [0],  [1],  [1].
%o A382923 (PARI)
%o A382923 G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
%o A382923 G_tx(10)
%Y A382923 Cf. A038207, A101509 (column k=0), A181331, A261780, A323429, A382924 (main diagonal).
%K A382923 nonn,easy,tabl
%O A382923 0,5
%A A382923 _John Tyler Rascoe_, Apr 09 2025