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A383713 Triangle read by rows: T(n,k) is the number of compositions of n with k parts all in standard order.

%I A383713 #10 May 07 2025 11:18:39
%S A383713 1,0,1,0,0,1,0,0,1,1,0,0,0,2,1,0,0,0,1,3,1,0,0,0,1,3,4,1,0,0,0,0,4,6,
%T A383713 5,1,0,0,0,0,2,10,10,6,1,0,0,0,0,1,9,20,15,7,1,0,0,0,0,1,7,25,35,21,8,
%U A383713 1,0,0,0,0,0,7,26,55,56,28,9,1,0,0,0,0,0,4,29,71,105,84,36,10,1
%N A383713 Triangle read by rows: T(n,k) is the number of compositions of n with k parts all in standard order.
%C A383713 A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.
%F A383713 G.f.: 1 + Sum_{i>0} y^i * x^(i*(i+1)/2) / Product_{j=1..i} 1 - y*(x - x^(j+1))/(1 - x).
%e A383713 Triangle begins:
%e A383713      k=0  1  2  3  4   5   6   7   8  9 10
%e A383713  n=0  [1],
%e A383713  n=1  [0, 1],
%e A383713  n=2  [0, 0, 1],
%e A383713  n=3  [0, 0, 1, 1],
%e A383713  n=4  [0, 0, 0, 2, 1],
%e A383713  n=5  [0, 0, 0, 1, 3,  1],
%e A383713  n=6  [0, 0, 0, 1, 3,  4,  1],
%e A383713  n=7  [0, 0, 0, 0, 4,  6,  5,  1],
%e A383713  n=8  [0, 0, 0, 0, 2, 10, 10,  6,  1],
%e A383713  n=9  [0, 0, 0, 0, 1,  9, 20, 15,  7, 1],
%e A383713  n=10 [0, 0, 0, 0, 1,  7, 25, 35, 21, 8, 1],
%e A383713  ...
%e A383713 Row n = 6 counts:
%e A383713  T(6,3) = 1: (1,2,3).
%e A383713  T(6,4) = 3: (1,1,2,2), (1,2,1,2), (1,2,2,1).
%e A383713  T(6,5) = 4: (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,2,1,1,1).
%e A383713  T(6,6) = 1: (1,1,1,1,1,1).
%o A383713 (PARI)
%o A383713 T_xy(max_row) = {my(N = max_row+1, x='x+O('x^N), h = 1 + sum(i=1,1+(N/2), y^i * x^(i*(i+1)/2)/prod(j=1,i, 1 - y*(x-x^(j+1))/(1-x)))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
%o A383713 T_xy(10)
%Y A383713 Cf. A000110 (column sums), A047998, A107429, A126347 (triangle transposed with no zeros), A278984, A383253 (row sums).
%K A383713 nonn,easy,tabl
%O A383713 0,14
%A A383713 _John Tyler Rascoe_, May 06 2025