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A384993 Triangle read by rows: T(n,k) is the number of compositions (p_0,p_1,...,p_m) of n such that max(i + p_i) = k for 0 <= i <= m.

%I A384993 #41 Sep 09 2025 16:03:31
%S A384993 1,0,1,0,0,2,0,0,1,3,0,0,0,4,4,0,0,0,3,8,5,0,0,0,1,12,13,6,0,0,0,0,12,
%T A384993 26,19,7,0,0,0,0,8,40,46,26,8,0,0,0,0,4,48,88,73,34,9,0,0,0,0,1,47,
%U A384993 140,163,108,43,10,0,0,0,0,0,38,190,307,273,152,53,11
%N A384993 Triangle read by rows: T(n,k) is the number of compositions (p_0,p_1,...,p_m) of n such that max(i + p_i) = k for 0 <= i <= m.
%e A384993 Triangle begins:
%e A384993     k=0  1  2  3   4   5   6   7  8
%e A384993  n=0 [1]
%e A384993  n=1 [0, 1]
%e A384993  n=2 [0, 0, 2]
%e A384993  n=3 [0, 0, 1, 3]
%e A384993  n=4 [0, 0, 0, 4,  4]
%e A384993  n=5 [0, 0, 0, 3,  8,  5]
%e A384993  n=6 [0, 0, 0, 1, 12, 13,  6]
%e A384993  n=7 [0, 0, 0, 0, 12, 26, 19,  7]
%e A384993  n=8 [0, 0, 0, 0,  8, 40, 46, 26, 8]
%e A384993 ...
%e A384993 The composition of n = 8, (2,1,3,1,1) has values of i + p_i : 2,2,5,4,5 that have a maximum value of 5 so this composition is counted under T(8,5) = 40.
%e A384993 T(4,3) = 4 counts: (1,2,1), (2,1,1), (2,2), (3,1).
%e A384993 T(4,4) = 4 counts: (1,1,1,1), (1,1,2), (1,3), (4).
%o A384993 (PARI)
%o A384993 r(i) = {sum(k=1,i, x^k)}
%o A384993 P(n) = {1 + sum(i=1,n, prod(j=0,i-1, r(n-j)))}
%o A384993 C(k) = {sum(i=1,k, x^k * P(k-i) * prod(j=0,i-2, 1 + r(k-j-2)))}
%o A384993 T_rowlist(max_row) = {my(N = max_row+1, h = 1 + sum(k=1,N, y^k * C(k))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
%Y A384993 Cf. A008930, A238859.
%Y A384993 Row sums A011782, empirical column sums A001339, empirical second diagonal A034856.
%K A384993 nonn,easy,tabl,new
%O A384993 0,6
%A A384993 _John Tyler Rascoe_, Sep 01 2025