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

0,4

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.

Links

Table of n, a(n) for n=0..54.

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.

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

Example

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

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 *)

Crossrefs

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

Diagonals: A239204, A000012, A016777.

Cf. A122538, A007318, A321620.

Sequence in context: A380113 A138263 A147721 * A114608 A154602 A216154

Adjacent sequences: A172091 A172092 A172093 * A172095 A172096 A172097

Keyword

nonn,tabl,changed

Author

Philippe Deléham, Jan 25 2010

Extensions

New name by Peter Luschny, Nov 27 2018

Status

approved