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 A110098 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y=x+1 to the line y = x). 2
 1, 2, 1, 6, 6, 1, 22, 30, 10, 1, 90, 146, 70, 14, 1, 394, 714, 430, 126, 18, 1, 1806, 3534, 2490, 938, 198, 22, 1, 8558, 17718, 14002, 6314, 1734, 286, 26, 1, 41586, 89898, 77550, 40054, 13338, 2882, 390, 30, 1, 206098, 461010, 426150, 244790, 94554 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1). The row sums are the central Delannoy numbers (A001850). Column 0 yields the large Schroeder numbers (A006318). Column 1 yields A006320. Column k has g.f. z^k*R^(2*k+1), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318). LINKS T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p 37. Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5. FORMULA T(n,k) = ((2*k+1)/(n-k))*Sum_{j=0..n-k} binomial(n-k, j)*binomial(n+k+j, n-k-1) for k < n; T(n,n) = 1; T(n,k) = 0 for k > n. G.f.: R/(1 - t*z*R^2), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318). Sum_{k=0..n} k*T(n,k) = A110099(n). T(n,k) = A033877(n-k+1, n+k+1). - Johannes W. Meijer, Sep 05 2013 EXAMPLE T(2, 1) = 6 because we have DN(E), N(E)D, N(E)EN, ND(E), NNE(E) and ENN(E) (the return E steps are shown between parentheses). Triangle begins:    1;    2,   1;    6,   6,   1;   22,  30,  10,   1;   90, 146,  70,  14,   1; MAPLE T := proc(n, k) if k=n then 1 else ((2*k+1)/(n-k))*sum(binomial(n-k, j)*binomial(n+k+j, n-k-1), j=0..n-k) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form CROSSREFS Cf. A001850, A006318, A006320, A033877, A110099, A110107. Sequence in context: A060538 A260848 A110183 * A244888 A130561 A157400 Adjacent sequences:  A110095 A110096 A110097 * A110099 A110100 A110101 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Jul 11 2005 STATUS approved

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Last modified April 19 21:57 EDT 2021. Contains 343117 sequences. (Running on oeis4.)