The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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(1-2*x-3*x^2))/(2*x*(1-x)), (1+x-sqrt(1-2*x-3*x^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 G. C. Greubel, Rows n = 0..100 of triangle, flattened Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019. I. Dolinka, J. East, R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 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>=1} T(n-1,k+j) with T(0,0)=1. - Philippe Deléham, Jan 23 2010 T(n,k) = (k/n)*Sum_{j=k..n} (-1)^(n-j)*C(n,j)*C(2*j-k-1,j-1), 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 *) PROG (PARI) T(n, k) = ((k+1)/(n+1))*sum(j=k+1, n+1, (-1)^(n-j+1)*binomial(n+1, j)* binomial(2*j-k-2, j-1) ); \\ G. C. Greubel, Feb 18 2020 (MAGMA) [((k+1)/(n+1))*(&+[(-1)^(n-j+1)*Binomial(n+1, j)*Binomial(2*j-k-2, j-1): j in [k+1..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020 (Sage) [[((k+1)/(n+1))*sum( (-1)^(n-j+1)*binomial(n+1, j)* binomial(2*j-k-2, j-1) for j in (k+1..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020 CROSSREFS Cf. A001006, A005043, A187306. Sequence in context: A091889 A147785 A067591 * A266692 A077884 A331103 Adjacent sequences:  A097606 A097607 A097608 * A097610 A097611 A097612 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 30 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 23 04:53 EDT 2020. Contains 337962 sequences. (Running on oeis4.)