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 A101275 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n having exactly k down steps hitting the x-axis. 4
 1, 1, 1, 1, 4, 1, 1, 13, 7, 1, 1, 44, 34, 10, 1, 1, 165, 150, 64, 13, 1, 1, 680, 659, 346, 103, 16, 1, 1, 3001, 2973, 1753, 659, 151, 19, 1, 1, 13880, 13844, 8716, 3798, 1116, 208, 22, 1, 1, 66345, 66300, 43384, 20798, 7226, 1744, 274, 25, 1, 1, 324908, 324853, 217804 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA G.f.: 2/[2-2z-t+tz+t*sqrt(1-6z+z^2)]. 1/(1-x-xy/(1-x-x/(1-x-x/(1-x-x/(1-x-x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009 T(n,k)= k*Sum_{m=0..n-k}(Sum_{i=0..m}(C(m+k,i)*C(2*m+k-i-1,m+k-1))*C(n-m,k))/(m+k)), T(n,0)=1. - Vladimir Kruchinin, Apr 20 2015 EXAMPLE Example. T(2,1)=4 because we have UHD, UUDD, HUD and UDH. Triangle begins: 1; 1,  1; 1,  4,  1; 1, 13,  7,  1; 1, 44, 34, 10, 1; MAPLE G:=2/(2-2*z-t+t*z+t*sqrt(1-6*z+z^2)): Gser:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form PROG (Maxima) T(n, k):=if k=0 then 1 else k*sum(((sum(binomial(m+k, i)*binomial(2*m+k-i-1, m+k-1), i, 0, m))*binomial(n-m, k))/(m+k), m, 0, n-k); /* Vladimir Kruchinin, Apr 20 2015 */ (PARI) T(n, k)= if (k==0, 1, k*sum(m=0, n-k, sum(i=0, m, binomial(m+k, i)*binomial(2*m+k-i-1, m+k-1)*binomial(n-m, k))/(m+k))); tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 21 2015 CROSSREFS Cf. A006318. Sequence in context: A227203 A140070 A158815 * A262494 A039755 A247502 Adjacent sequences:  A101272 A101273 A101274 * A101276 A101277 A101278 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Dec 20 2004 STATUS approved

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Last modified May 13 01:45 EDT 2021. Contains 343830 sequences. (Running on oeis4.)