This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A114709 #32 May 05 2025 03:53:34
%S A114709 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 A114709 2502,1192,414,128,30,12,0,1,12194,5772,2006,600,170,36,14,0,1,60570,
%U A114709 28536,9882,2976,810,216,42,16,0,1,305526,143388,49554,14904,4110,1044
%N A114709 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 A114709 Row sums are the little Schroeder numbers (A001003). Column 0 is A114710. Sum_{k=0..n} k*T(n,k) = A010683(n-1).
%C A114709 Riordan array ((1+3*x-sqrt(1-6*x+x^2))/(2*x*(2*x+3)),(1+3*x-sqrt(1-6*x+x^2))/(2*(2*x+3))), inverse of the Riordan array ((1-3*x)/((1-x)*(1-2*x)), x*(1-3*x)/((1-x)*(1-2*x))). - _Paul Barry_, Mar 01 2011
%H A114709 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 A114709 Emeric 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 A114709 Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%F A114709 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 A114709 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 A114709 T(4,2)=6 because we have (HH)UHD,(HH)UUDD,(H)UHD(H),(H)UUDD(H),UHD(HH) and 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 A114709 Triangle starts:
%e A114709 1;
%e A114709 0,1;
%e A114709 2,0,1;
%e A114709 6,4,0,1;
%e A114709 26,12,6,0,1;
%e A114709 Production matrix is
%e A114709 0, 1,
%e A114709 2, 0, 1,
%e A114709 6, 2, 0, 1,
%e A114709 18, 6, 2, 0, 1,
%e A114709 54, 18, 6, 2, 0, 1,
%e A114709 162, 54, 18, 6, 2, 0, 1,
%e A114709 486, 162, 54, 18, 6, 2, 0, 1,
%e A114709 1458, 486, 162, 54, 18, 6, 2, 0, 1
%e A114709 where the columns have generator (1-x)*(1-2*x)/(1-3*x).
%p A114709 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 A114709 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 A114709 (Maxima)
%o A114709 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 A114709 Cf. A001003, A114710, A010683.
%K A114709 nonn,tabl
%O A114709 0,4
%A A114709 _Emeric Deutsch_, Dec 26 2005