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A101275
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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.
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4
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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
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OFFSET
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0,5
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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