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

0,4

Comments

Reverse of A144944. Inverse of A186827.

Links

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

Jorge Ballarín, Jorge Delgado, and Juan Manuel Peña, Accurate computations with Riordan arrays associated with Schröder matrices, Calcolo 63, 13 (2026). See pp. 9-10.

Formula

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

Example

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.

Mathematica

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 *)

Prog

(Haskell)
a186826 n k = a186826_tabl !! n !! k
a186826_row n = a186826_tabl !! n
a186826_tabl = map reverse a144944_tabl
-- Reinhard Zumkeller, May 11 2013
(SageMath)
@CachedFunction
def t(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (k<n-1): return t(n, k-1) + t(n-1, k) + t(n-1, k-1)
    else: return -t(n, n-2)
def A186826(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

Crossrefs

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

Sequence in context: A131889 A292386 A174287 * A185418 A050609 A120870

Adjacent sequences: A186823 A186824 A186825 * A186827 A186828 A186829

Keyword

nonn,easy,tabl

Author

Paul Barry, Feb 27 2011

Status

approved