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%I #35 Nov 27 2018 19:27:16
%S 1,1,1,3,4,1,11,17,7,1,45,76,40,10,1,197,353,216,72,13,1,903,1688,
%T 1145,458,113,16,1,4279,8257,6039,2745,829,163,19,1,20793,41128,31864,
%U 15932,5558,1356,222,22,1,103049,207905,168584,90776,35318,10070,2066,290,25,1
%N The Riordan square of the little Schröder numbers A001003.
%C The Riordan square is defined in A321620.
%C Previous name was: Triangle, read by rows, given by [1,2,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.
%H P. Barry, <a href="http://arxiv.org/abs/1311.7161">Comparing two matrices of generalized moments defined by continued fraction expansions</a>, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry3/barry291.html">J. Int. Seq. 17 (2014) # 14.5.1</a>.
%H Johann Cigler, <a href="https://arxiv.org/abs/1611.05252">Some elementary observations on Narayana polynomials and related topics</a>, arXiv:1611.05252 [math.CO], 2016. See p. 12.
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://arxiv.org/abs/math/0702638">Production Matrices and Riordan arrays</a>, arXiv:math/0702638 [math.CO], 2007.
%H Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.
%F T(0, 0) = 1, T(n, k) = 0 if k>n, T(n, 0) = T(n-1, 0) + 2*T(n-1, 1), T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + 2*T(n-1, k+1) for k>0.
%F Sum_{0<=k<=n} T(n, k) = A109980(n).
%F Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).
%e Triangle begins:
%e 1
%e 1, 1
%e 3, 4, 1
%e 11, 17, 7, 1
%e 45, 76, 40, 10, 1
%e 197, 353, 216, 72, 13, 1
%e 903, 1688, 1345, 458, 113, 16, 1
%e 4279, 8257, 6039, 2745, 829, 163, 19, 1
%e 20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1
%e 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
%e .
%e Production matrix begins:
%e 1, 1
%e 2, 3, 1
%e 0, 2, 3, 1
%e 0, 0, 2, 3, 1
%e 0, 0, 0, 2, 3, 1
%e 0, 0, 0, 0, 2, 3, 1
%e 0, 0, 0, 0, 0, 2, 3, 1
%e 0, 0, 0, 0, 0, 0, 2, 3, 1
%e ... - _Philippe Deléham_, Sep 24 2014
%t DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k+1]] + y s[[k+1]]; p[0, _] = 1; p[_, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
%t nmax = 9;
%t DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%t (* Function RiordanSquare defined in A321620. *)
%t RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten (* _Peter Luschny_, Nov 27 2018 *)
%Y T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).
%Y Diagonals: A239204, A000012, A016777.
%Y Cf. A122538, A007318, A321620.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Jan 25 2010
%E New name by _Peter Luschny_, Nov 27 2018
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