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A137680 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.

%I A137680 #13 Apr 02 2024 11:35:00
%S A137680 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 A137680 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 A137680 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 A137680 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 A137680 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 A137680 Left column starting (1, 3, ...) = INVERT transform of A160096. - _Gary W. Adamson_, May 01 2009
%e A137680 First few rows of the triangle:
%e A137680      1;
%e A137680      1,   1;
%e A137680      3,   0,   1;
%e A137680      7,   2,   0,   1;
%e A137680     17,   4,   1,   0,  1;
%e A137680     40,  10,   4,   1,  0,  1;
%e A137680     96,  23,   8,   3,  1,  0, 1;
%e A137680    228,  56,  19,   8,  3,  1, 0, 1;
%e A137680    544, 132,  46,  18,  7,  3, 1, 0, 1;
%e A137680   1296, 316, 109,  42, 18,  7, 3, 1, 0, 1;
%e A137680   3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
%e A137680   ...
%p A137680 A137680 := proc(n,k)
%p A137680     if k < 1 or k > n then
%p A137680         0 ;
%p A137680     elif n = 1 then
%p A137680         1;
%p A137680     elif k = 1 then
%p A137680         add(add(procname(r,j),j=1..r),r=1..n-1) ;
%p A137680     else
%p A137680         procname(n-1,k-1)-procname(n-k,k-1) ;
%p A137680     end if;
%p A137680 end proc: # _R. J. Mathar_, Aug 12 2012
%t A137680 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 A137680 Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 02 2024, after _R. J. Mathar_ *)
%Y A137680 Cf. A137681 (row sums), A137682.
%Y A137680 Cf. A160096. - _Gary W. Adamson_, May 01 2009
%K A137680 nonn,tabl
%O A137680 1,4
%A A137680 _Gary W. Adamson_, Feb 05 2008