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%I #13 Apr 02 2024 11:35:00
%S 1,1,1,3,0,1,7,2,0,1,17,4,1,0,1,40,10,4,1,0,1,96,23,8,3,1,0,1,228,56,
%T 19,8,3,1,0,1,544,132,46,18,7,3,1,0,1,1296,316,109,42,18,7,3,1,0,1,
%U 3089,752,260,101,41,17,7,3,1,0,1,7361,1793,620,241,98,41,17,7,3,1,0,1,17544,4272,1477,574,233,97,40,17,7,3,1,0,1
%N Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.
%C A variation of the same sequence = column 2 of the triangle: (1, 0, 2, 4, 10, 23, 56, 132, ...) = first difference row of column 1. Left border of the triangle = A137682.
%C Left column starting (1, 3, ...) = INVERT transform of A160096. - _Gary W. Adamson_, May 01 2009
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 3, 0, 1;
%e 7, 2, 0, 1;
%e 17, 4, 1, 0, 1;
%e 40, 10, 4, 1, 0, 1;
%e 96, 23, 8, 3, 1, 0, 1;
%e 228, 56, 19, 8, 3, 1, 0, 1;
%e 544, 132, 46, 18, 7, 3, 1, 0, 1;
%e 1296, 316, 109, 42, 18, 7, 3, 1, 0, 1;
%e 3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
%e ...
%p A137680 := proc(n,k)
%p if k < 1 or k > n then
%p 0 ;
%p elif n = 1 then
%p 1;
%p elif k = 1 then
%p add(add(procname(r,j),j=1..r),r=1..n-1) ;
%p else
%p procname(n-1,k-1)-procname(n-k,k-1) ;
%p end if;
%p end proc: # _R. J. Mathar_, Aug 12 2012
%t T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
%t Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 02 2024, after _R. J. Mathar_ *)
%Y Cf. A137681 (row sums), A137682.
%Y Cf. A160096. - _Gary W. Adamson_, May 01 2009
%K nonn,tabl
%O 1,4
%A _Gary W. Adamson_, Feb 05 2008
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