%I #28 Jul 24 2017 11:56:12
%S 0,0,0,1,0,0,2,1,0,0,5,3,1,0,0,10,9,4,1,0,0,20,24,14,5,1,0,0,38,64,44,
%T 20,6,1,0,0,71,173,135,71,27,7,1,0,0,130,485,414,241,106,35,8,1,0,0,
%U 235,1420,1290,805,391,150,44,9,1,0,0
%N Riordan array (A000045(x)^m, x*A000108(x)), m = 2.
%H G. C. Greubel, <a href="/A185663/b185663.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2013.
%F R(n,k,m) = k*Sum_{i=0..n-k} (Sum_{j=ceiling((i-m)/2)..i-m} binomial(j, i-m-j) * binomial(m+j-1, m-1)) * binomial(2*(n-i)-k-1, n-i-1)/(n-i), k > 0.
%F R(n,0,m) = Sum_{j=ceiling((n-m)/2)..n-m} binomial(j,n-m-j) * binomial(m+j-1,m-1), m = 2.
%e Array begins
%e 0;
%e 0, 0;
%e 1, 0, 0;
%e 2, 1, 0, 0;
%e 5, 3, 1, 0, 0;
%e 10, 9, 4, 1, 0, 0;
%e 20, 24, 14, 5, 1, 0, 0;
%e 38, 64, 44, 20, 6, 1, 0, 0;
%t R[n_, k_, m_] := k*Sum[Sum[Binomial[j, i - m - j]*Binomial[m - 1 + j, m - 1], {j, Ceiling[(i - m)/2], i - m}]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k} ]; R[n_, 0, m_] := Sum[Binomial[j, n - m - j]*Binomial[j + m - 1, m - 1], {j, Ceiling[(n - m)/2], n - m}];
%t Table[R[n, k, 2], {n, 0, 10}, {k, 0, n}] (* _G. C. Greubel_, Jul 23 2017 *)
%Y Cf. A185937, A139375 (m = 1).
%K nonn,tabl
%O 0,7
%A _Vladimir Kruchinin_, Feb 08 2011