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%I #19 Sep 16 2019 15:37:57
%S 1,1,1,1,1,1,1,2,3,2,1,3,6,7,5,1,4,10,16,18,12,1,5,15,30,44,46,30,1,6,
%T 21,50,89,120,120,76,1,7,28,77,160,259,329,316,196,1,8,36,112,265,496,
%U 748,904,841,512,1,9,45,156,413,873,1509,2148,2493,2257,1353,1,10,55,210,614,1442,2795,4530,6150,6898,6103,3610
%N Triangle T read by rows: differences of Motzkin triangle (A026300).
%C For n >= 2, T(n,k)= number of nonnegative integer strings s(0),...,s(n) such that s(n)=n-k, s(0)=s(1)=1, |s(i)-s(i-1)|<=1 for i >= 2.
%F T(n, k) = A026300(n, k) - A026300(n-1, k-1), T(1, 1) = 1.
%F T(i, 0)=1 for i >= 0, T(2, 1)=1, T(2, 2)=1, T(3, 1)=2, T(3, 2)=3, T(3, 3)=2; and for i >= 4, T(i, 1)=i-1, T(i, i)=T(i-1, i-2)+T(i-1, i-1) and T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i-1.
%F Right-hand columns have g.f. (1-z)*M^k, where M is g.f. of Motzkin numbers (A001006).
%e 1
%e 1,1
%e 1,1,1
%e 1,2,3,2
%e 1,3,6,7,5
%e 1,4,10,16,18,12
%e 1,5,15,30,44,46,30
%Y Right-hand columns include A002026, A026107, A026134, A026109, A026110.
%Y Row sums are in A025566. Central column is in A026112.
%K nonn,tabl
%O 0,8
%A _Clark Kimberling_
%E Edited by _Ralf Stephan_, Dec 18 2004
%E a(65) corrected and more terms from _Sean A. Irvine_, Sep 16 2019
%E Offset set to 0 by _Alois P. Heinz_, Sep 16 2019
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