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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. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Reverse of A144944. Inverse of A186827. Row sums are A010683. Diagonal sums are A186828. LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened FORMULA Riordan array ((1+x+sqrt(1-6x+x^2))/(4x), (1-x-sqrt(1-6x+x^2))/2). 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). [From Vladimir Kruchinin, Mar 09 2011] 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 CROSSREFS 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

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)