A172094 The Riordan square of the little Schröder numbers A001003.

1, 1, 1, 3, 4, 1, 11, 17, 7, 1, 45, 76, 40, 10, 1, 197, 353, 216, 72, 13, 1, 903, 1688, 1145, 458, 113, 16, 1, 4279, 8257, 6039, 2745, 829, 163, 19, 1, 20793, 41128, 31864, 15932, 5558, 1356, 222, 22, 1, 103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1

Offset: 0

Author

Philippe Deléham, Jan 25 2010

Keywords

• nonn
• tabl

Comments

The Riordan square is defined in A321620.

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.

Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.

Examples

			Triangle begins:
     1
     1,      1
     3,      4,      1
    11,     17,      7,     1
    45,     76,     40,    10,     1
   197,    353,    216,    72,    13,     1
   903,   1688,   1345,   458,   113,    16,    1
  4279,   8257,   6039,  2745,   829,   163,   19,   1
20793,  41128,  31864, 15932,  5558,  1356,  222,  22,  1
103049, 207905, 168584, 90776, 35318, 10070, 2066, 290, 25, 1
.
Production matrix begins:
1, 1
2, 3, 1
0, 2, 3, 1
0, 0, 2, 3, 1
0, 0, 0, 2, 3, 1
0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 2, 3, 1
0, 0, 0, 0, 0, 0, 2, 3, 1
... - _Philippe Deléham_, Sep 24 2014
		

Links

• P. Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
• Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 12.
• E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
• Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Crossrefs

T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).

Diagonals: A239204, A000012, A016777.

Cf. A122538, A007318, A321620.

Programs

Maple

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):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jan 24 2025

Mathematica

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}]];
nmax = 9;
DELTA[Table[{1, 2}, (nmax+1)/2] // Flatten, Prepend[Table[0, {nmax}], 1], nmax] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[(1 + x - Sqrt[1 - 6x + x^2])/(4x), 11] // Flatten  (* Peter Luschny, Nov 27 2018 *)

Formula

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.

Sum_{0<=k<=n} T(n, k) = A109980(n).

Sum_{k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n).

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

Extensions

New name by Peter Luschny, Nov 27 2018