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A172094 The Riordan square of the little Schröder numbers A001003.

%I A172094 #37 Jan 24 2025 16:36:51
%S A172094 1,1,1,3,4,1,11,17,7,1,45,76,40,10,1,197,353,216,72,13,1,903,1688,
%T A172094 1145,458,113,16,1,4279,8257,6039,2745,829,163,19,1,20793,41128,31864,
%U A172094 15932,5558,1356,222,22,1,103049,207905,168584,90776,35318,10070,2066,290,25,1
%N A172094 The Riordan square of the little Schröder numbers A001003.
%C A172094 The Riordan square is defined in A321620.
%C A172094 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 A172094 Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.
%H A172094 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 A172094 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 A172094 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 A172094 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 A172094 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 A172094 Sum_{0<=k<=n} T(n, k) = A109980(n).
%F A172094 Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).
%F A172094 T(n, k) = Sum_{j=0..n-k} (binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j. (Cigler) - _Peter Luschny_, Jan 24 2025
%e A172094 Triangle begins:
%e A172094      1
%e A172094      1,      1
%e A172094      3,      4,      1
%e A172094     11,     17,      7,     1
%e A172094     45,     76,     40,    10,     1
%e A172094    197,    353,    216,    72,    13,     1
%e A172094    903,   1688,   1345,   458,   113,    16,    1
%e A172094   4279,   8257,   6039,  2745,   829,   163,   19,   1
%e A172094 20793,  41128,  31864, 15932,  5558,  1356,  222,  22,  1
%e A172094 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
%e A172094 .
%e A172094 Production matrix begins:
%e A172094 1, 1
%e A172094 2, 3, 1
%e A172094 0, 2, 3, 1
%e A172094 0, 0, 2, 3, 1
%e A172094 0, 0, 0, 2, 3, 1
%e A172094 0, 0, 0, 0, 2, 3, 1
%e A172094 0, 0, 0, 0, 0, 2, 3, 1
%e A172094 0, 0, 0, 0, 0, 0, 2, 3, 1
%e A172094 ... - _Philippe Deléham_, Sep 24 2014
%p A172094 T := (n, k) -> local j; add((binomial(n-1, j)*binomial(n+1, k+j+1) - binomial(n, j)*binomial(n, k+j+1))*2^j, j = 0..n-k):
%p A172094 for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # _Peter Luschny_, Jan 24 2025
%t A172094 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 A172094 nmax = 9;
%t A172094 DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%t A172094 (* Function RiordanSquare defined in A321620. *)
%t A172094 RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten  (* _Peter Luschny_, Nov 27 2018 *)
%Y A172094 T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).
%Y A172094 Diagonals: A239204, A000012, A016777.
%Y A172094 Cf. A122538, A007318, A321620.
%K A172094 nonn,tabl
%O A172094 0,4
%A A172094 _Philippe Deléham_, Jan 25 2010
%E A172094 New name by _Peter Luschny_, Nov 27 2018