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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.

%I A186826 #19 Mar 11 2023 08:13:59
%S A186826 1,1,1,3,3,1,11,11,5,1,45,45,23,7,1,197,197,107,39,9,1,903,903,509,
%T A186826 205,59,11,1,4279,4279,2473,1061,347,83,13,1,20793,20793,12235,5483,
%U A186826 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
%N 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.
%C A186826 Reverse of A144944. Inverse of A186827.
%H A186826 Reinhard Zumkeller, <a href="/A186826/b186826.txt">Rows n = 0..125 of triangle, flattened</a>
%F A186826 Riordan array ((1+x+sqrt(1-6*x+x^2))/(4*x), (1-x-sqrt(1-6*x+x^2))/2).
%F A186826 Sum_{k=0..n} T(n,k) = A010683(n).
%F A186826 Sum_{k=0..floor(n/2)} T(n-k, k) = A186828(n).
%F A186826 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
%F A186826 Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2). - _G. C. Greubel_, Mar 11 2023
%e A186826 Triangle begins
%e A186826        1;
%e A186826        1,      1;
%e A186826        3,      3,      1;
%e A186826       11,     11,      5,      1;
%e A186826       45,     45,     23,      7,     1;
%e A186826      197,    197,    107,     39,     9,     1;
%e A186826      903,    903,    509,    205,    59,    11,    1;
%e A186826     4279,   4279,   2473,   1061,   347,    83,   13,    1;
%e A186826    20793,  20793,  12235,   5483,  1949,   541,  111,   15,   1;
%e A186826   103049, 103049,  61463,  28435, 10717,  3285,  795,  143,  17,  1;
%e A186826   518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1;
%e A186826 Production matrix of this triangle begins
%e A186826   1, 1;
%e A186826   2, 2, 1;
%e A186826   2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 2, 2, 2, 2, 1;
%e A186826   2, 2, 2, 2, 2, 2, 2, 2, 2, 1;
%e A186826 For instance, 107=1*45+2*23+2*7+2*1.
%t A186826 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];
%t A186826 Table[t[p, q], {p,0,10}, {q,p,0,-1}]//Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%o A186826 (Haskell)
%o A186826 a186826 n k = a186826_tabl !! n !! k
%o A186826 a186826_row n = a186826_tabl !! n
%o A186826 a186826_tabl = map reverse a144944_tabl
%o A186826 -- _Reinhard Zumkeller_, May 11 2013
%o A186826 (SageMath)
%o A186826 @CachedFunction
%o A186826 def t(n,k):
%o A186826     if (k<0 or k>n): return 0
%o A186826     elif (k==0): return 1
%o A186826     elif (k<n-1): return t(n,k-1) + t(n-1,k) + t(n-1,k-1)
%o A186826     else: return -t(n,n-2)
%o A186826 def A186826(n,k): return t(n+2,n-k)
%o A186826 flatten([[A186826(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 11 2023
%Y A186826 Cf. A001003, A006318, A010683 (row sums), A144944 (row reverse), A186827 (inverse), A186828 (diagonal sums), A239204.
%K A186826 nonn,easy,tabl
%O A186826 0,4
%A A186826 _Paul Barry_, Feb 27 2011