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 A026120 Triangle T by rows: second differences of Motzkin triangle (A026300), (i >= -1, -1<=j<=i). 22
 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 4, 4, 3, 1, 3, 7, 10, 11, 7, 1, 4, 11, 20, 28, 28, 18, 1, 5, 16, 35, 59, 76, 74, 46, 1, 6, 22, 56, 110, 170, 209, 196, 120, 1, 7, 29, 84, 188, 336, 489, 575, 525, 316, 1, 8, 37, 120, 301, 608, 1013, 1400, 1589, 1416, 841, 1, 9, 46, 165, 458, 1029, 1922, 3021, 4002, 4405, 3846, 2257 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET -1,9 COMMENTS For n >= 2, T(n,k) = number of nonnegative integer strings s(0),...,s(n) such that s(0)=1, s(n)=n-k, |s(1)-1|=1, |s(i)-s(i-1)|<=1 for i >= 2. LINKS FORMULA T(n, k) = A026105(n+1, k+1) - A026105(n, k), T(0, 0) = 0. T(i, 0)=1 for i >= -1; T(0, 1)=0; T(1, 1)=1, T(1, 2)=0; for i >= 2, T(i, 1)=i-1, T(i, i+1)=T(i-1, i-1)+T(i-1, i), T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i. G.f. of right-hand columns is (1-z)^2*M^k, with M the g.f. of the Motzkin numbers (A001006). EXAMPLE 1; 1,0; 1,1,0; 1,1,2,1; 1,2,4,4,3; 1,3,7,10,11,7; ... PROG (PARI) T(i, j)=if(j<0||j>i+1, 0, if(j==0, 1, if(j==1, if(i>1, i-1, i>0), if(i+1==j, if(i==1, 0, T(i-1, i-1)+T(i-1, i)), T(i-1, j-2)+T(i-1, j-1)+T(i-1, j))))) \\ Ralf Stephan CROSSREFS Cf. Right-hand columns include A026107, A026122, A026123, A026124, A026125, A026126. Row sums are A026135. Central column is A026127. Sequence in context: A085143 A321029 A253473 * A108746 A333270 A333272 Adjacent sequences:  A026117 A026118 A026119 * A026121 A026122 A026123 KEYWORD nonn,tabl AUTHOR EXTENSIONS Edited by Ralf Stephan, Dec 18 2004 a(26) corrected and more terms from Sean A. Irvine, Sep 17 2019 STATUS approved

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Last modified September 24 12:20 EDT 2020. Contains 337318 sequences. (Running on oeis4.)