%I #22 Oct 30 2019 22:06:04
%S 1,0,1,2,0,1,6,4,0,1,26,12,6,0,1,114,56,18,8,0,1,526,252,90,24,10,0,1,
%T 2502,1192,414,128,30,12,0,1,12194,5772,2006,600,170,36,14,0,1,60570,
%U 28536,9882,2976,810,216,42,16,0,1,305526,143388,49554,14904,4110,1044
%N Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k horizontal steps on the x-axis (0<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.
%C Row sums are the little Schroeder numbers (A001003). Column 0 is A114710. Sum(k*T(n,k),k=0..n)=A010683(n-1).
%C Riordan array ((1+3x-sqrt(1-6x+x^2))/(2x(2x+3)),(1+3x-sqrt(1-6x+x^2))/(2(2x+3))), inverse of the Riordan array ((1-3x)/((1-x)(1-2x)), (x(1-3x)/((1-x)(1-2x))). - _Paul Barry_, Mar 01 2011
%H Michael De Vlieger, <a href="/A114709/b114709.txt">Table of n, a(n) for n = 0..11475</a> (assuming the Kruchinin formula, rows 0 <= n <= 150, flattened.)
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
%H Shishuo Fu, Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%F G.f.: 1/(1+z-t*z-z*R), where R=(1-z-sqrt(1-6*z+z^2))/(2*z) is the g.f. of the large Schroeder numbers (A006318).
%F T(n,k) = Sum_{i=0..n-k}((i+1)*binomial(k+i+1,k)*Sum_{j=0..n-k-i}((-1)^(j+i)*2^(n-k-j-i)*binomial(n+1,j)*binomial(2*n-k-j-i,n)))/(n+1). - _Vladimir Kruchinin_, Feb 29 2016
%e T(4,2)=6 because we have (HH)UHD,(HH)UUDD,(H)UHD(H),(H)UUDD(H),UHD(HH) and
%e UUDD(HH), where U=(1,1), D=(1,-1) and H=(2,0) (the H's on the x-axis are shown between parentheses).
%e Triangle starts:
%e 1;
%e 0,1;
%e 2,0,1;
%e 6,4,0,1;
%e 26,12,6,0,1;
%e Production matrix is
%e 0, 1,
%e 2, 0, 1,
%e 6, 2, 0, 1,
%e 18, 6, 2, 0, 1,
%e 54, 18, 6, 2, 0, 1,
%e 162, 54, 18, 6, 2, 0, 1,
%e 486, 162, 54, 18, 6, 2, 0, 1,
%e 1458, 486, 162, 54, 18, 6, 2, 0, 1
%e where the columns have generator ((1-x)(1-2x))/(1-3x).
%p R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1+z-t*z-z*R): Gser:=simplify(series(G,z=0,14)): 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^j),j=1..n+1) od; # yields sequence in triangular form
%t Table[Sum[(i + 1) Binomial[k + i + 1, k] Sum[(-1)^(j + i)*2^(n - k - j - i)* Binomial[n + 1, j] Binomial[2 n - k - j - i, n], {j, 0, n - k - i}], {i, 0, n - k}]/(n + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Oct 30 2019 *)
%o (Maxima)
%o T(n,k):=sum((i+1)*binomial(k+i+1,k)*sum((-1)^(j+i)*2^(n-k-j-i)*binomial(n+1,j)*binomial(2*n-k-j-i,n),j,0,n-k-i),i,0,n-k)/(n+1); /* _Vladimir Kruchinin_, Feb 29 2016 */
%Y Cf. A001003, A114710, A010683.
%K nonn,tabl
%O 0,4
%A _Emeric Deutsch_, Dec 26 2005