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 A097609 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0. 11
 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 2, 3, 0, 1, 6, 7, 3, 4, 0, 1, 15, 14, 12, 4, 5, 0, 1, 36, 37, 24, 18, 5, 6, 0, 1, 91, 90, 67, 36, 25, 6, 7, 0, 1, 232, 233, 165, 106, 50, 33, 7, 8, 0, 1, 603, 602, 438, 264, 155, 66, 42, 8, 9, 0, 1, 1585, 1586, 1147, 719, 390, 215, 84, 52, 9, 10, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Row sums give the Motzkin numbers (A001006). Column 0 is A005043. Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1-x)),(1+x-sqrt(1-2x-3x^2))/(2(1-x))). - Paul Barry, Jun 21 2008 Inverse of Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)), which is A104597. - Paul Barry, Jun 21 2008 Triangle read by rows, product of A064189 and A130595 considered as infinite lower triangular arrays; A097609 = A064189*A130195 = B*A053121*B^(-1) where B = A007318. - Philippe Deléham, Dec 07 2009 T(n+1,1) = A187306(n). - Philippe Deléham, Jan 28 2014 The number of lattice paths from (0,0) to (n,k) that do not cross below the x-axis and use up-step=(1,1) and down-steps=(1,-z) where z is a positive integer. For example, T(4,0) = 3: [(1,1)(1,1)(1,-1)(1,-1)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,1)(1,-3)]. - Nicholas Ham, Aug 20 2015 LINKS I Dolinka, J East, RD Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279, 2015. D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Trends in Mathematics 2000, pp 127-139. FORMULA G.f.: 2/(1-2*t*z+z+sqrt(1-2*z-3*z^2)). T(n,k) = T(n-1,k-1)+ Sum_{j, j>=1} T(n-1,k+j) ; T(0,0)=1 . [Philippe Deléham, Jan 23 2010] T(n,k) = k/n*sum(j=k..n, C(n,j)*C(2*j-k-1,j-1)*(-1)^(n-j)), n>0. - Vladimir Kruchinin, Feb 05 2011 EXAMPLE Triangle begins: 1; 0,1; 1,0,1; 1,2,0,1; 3,2,3,0,1; 6,7,3,4,0,1; Row n has n+1 terms. T(5,2) = 3 because (HH)UHD,(H)UHD(H) and UHD(HH) are the only Motzkin paths of length 5 with 2 horizontal steps at level 0 (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1). Production matrix begins 0, 1 1, 0, 1 1, 1, 0, 1 1, 1, 1, 0, 1 1, 1, 1, 1, 0, 1 1, 1, 1, 1, 1, 0, 1 1, 1, 1, 1, 1, 1, 0, 1 1, 1, 1, 1, 1, 1, 1, 0, 1 1, 1, 1, 1, 1, 1, 1, 1, 0, 1 ... - Philippe Deléham, Mar 02 2013 MAPLE G:=2/(1-2*t*z+z+sqrt(1-2*z-3*z^2)): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..12); MATHEMATICA nmax = 12; t[n_, k_] := ((-1)^(n+k)*k*n!*HypergeometricPFQ[{(k+1)/2, k/2, k-n}, {k, k+1}, 4])/(n*k!*(n-k)!); Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *) CROSSREFS Cf. A001006, A005043, A187306. Sequence in context: A091889 A147785 A067591 * A266692 A077884 A267724 Adjacent sequences:  A097606 A097607 A097608 * A097610 A097611 A097612 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 30 2004 STATUS approved

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Last modified December 16 01:32 EST 2019. Contains 330013 sequences. (Running on oeis4.)