A239204 -id:A239204 - cp's OEIS frontend

A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 45, 45, 23, 7, 1, 197, 197, 107, 39, 9, 1, 903, 903, 509, 205, 59, 11, 1, 4279, 4279, 2473, 1061, 347, 83, 13, 1, 20793, 20793, 12235, 5483, 1949, 541, 111, 15, 1, 103049, 103049, 61463, 28435, 10717, 3285, 795, 143, 17, 1, 518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1

Offset: 0

Paul Barry, Feb 27 2011

Reverse of A144944. Inverse of A186827.

			Triangle begins
       1;
       1,      1;
       3,      3,      1;
      11,     11,      5,      1;
      45,     45,     23,      7,     1;
     197,    197,    107,     39,     9,     1;
     903,    903,    509,    205,    59,    11,    1;
    4279,   4279,   2473,   1061,   347,    83,   13,    1;
   20793,  20793,  12235,   5483,  1949,   541,  111,   15,   1;
  103049, 103049,  61463,  28435, 10717,  3285,  795,  143,  17,  1;
  518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1;
Production matrix of this triangle begins
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 2, 1;
For instance, 107=1*45+2*23+2*7+2*1.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A001003, A006318, A010683 (row sums), A144944 (row reverse), A186827 (inverse), A186828 (diagonal sums), A239204.

a186826 n k = a186826_tabl !! n !! k
a186826_row n = a186826_tabl !! n
a186826_tabl = map reverse a144944_tabl
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q -1] + t[p-1, q] + t[p-1, q-1];
Table[t[p, q], {p,0,10}, {q,p,0,-1}]//Flatten (* Jean-François Alcover, Jul 16 2019 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kA186826(n,k): return t(n+2,n-k)
flatten([[A186826(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 11 2023

Riordan array ((1+x+sqrt(1-6*x+x^2))/(4*x), (1-x-sqrt(1-6*x+x^2))/2).
Sum_{k=0..n} T(n,k) = A010683(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A186828(n).
R(n,k) = k*Sum_{i=0..n-k} (A001003(i)/(n-i))*Sum_{m=0..n-k-i} binomial(n-i,m)*binomial(2*(n-i)-m-k-1, n-i-1), k>0, R(n,0) = A001003(n). - Vladimir Kruchinin, Mar 09 2011
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2). - G. C. Greubel, Mar 11 2023

A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 5, 11, 11, 1, 7, 23, 45, 45, 1, 9, 39, 107, 197, 197, 1, 11, 59, 205, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793, 1, 17, 143, 795, 3285, 10717, 28435, 61463, 103049, 103049

Offset: 0

Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008

			First few rows of the triangle:
  1
  1,  1
  1,  3,  3
  1,  5, 11,  11
  1,  7, 23,  45,  45
  1,  9, 39, 107, 197, 197
  1, 11, 59, 205, 509, 903, 903

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Johannes Fischer and Volker Heun, Finding Range Minima in the Middle: Approximations and Applications, Math. Comput. Sci. 3(1): 17-30 (2010). See Eq. (3.3)
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.

Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.
Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.
Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).
Cf. A010683 (row sums), A186826 (rows reversed).
Cf. A239204, A247623.

a144944 n k = a144944_tabl !! n !! k
a144944_row n = a144944_tabl !! n
a144944_tabl = iterate f [1] where
   f us = vs ++ [last vs] where
     vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q-1] + t[p-1, q] + t[p-1, q-1]; Flatten[Table[ t[p, q], {p,0,6}, {q,0, p}]] (* Jean-François Alcover, Dec 19 2011 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kG. C. Greubel, Mar 11 2023

From G. C. Greubel, Mar 11 2023: (Start)
Sum_{k=0..n} T(n, k) = A010683(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)

A172094 The Riordan square of the little Schröder numbers A001003.

Original entry on oeis.org

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

Philippe Deléham, Jan 25 2010

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.

			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

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.

T(n, 0) = A001003(n) (little Schröder), A109980 (row sums).
Diagonals: A239204, A000012, A016777.
Cf. A122538, A007318, A321620.

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

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 *)

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

New name by Peter Luschny, Nov 27 2018

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A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

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A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

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A172094 The Riordan square of the little Schröder numbers A001003.

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