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%I #20 Sep 06 2017 04:07:50
%S 1,2,1,10,4,1,66,24,6,1,498,172,42,8,1,4066,1360,326,64,10,1,34970,
%T 11444,2706,536,90,12,1,312066,100520,23526,4672,810,120,14,1,2862562,
%U 911068,211546,42024,7410,1156,154,16,1
%N A convolution triangle of numbers based on A027307.
%C Triangle T(n,k) for A(x)^k = Sum_{n>=k} T(n,k)*x^n, where o.g.f. A(x) satisfies A(x) = (1+x*A(x)^2)/(1-x*A(x)^2). - _Vladimir Kruchinin_, Mar 16 2011
%H G. C. Greubel, <a href="/A110682/b110682.txt">Table of n, a(n) for the first 100 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], 2011-2013.
%F T(0, 0) = 1; T(n, k) = 0 if k<0 or if k>n; T(n, k) = Sum_{j, j>=0} T(n-1, k-1+j)*A006318(j).
%F Sum_{k, k>=0} T(n, k) = A108442(n+1).
%F T(n,k) = k/(2*n-k)*Sum_{i=0,n-k} binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1), n >= k > 0. - _Vladimir Kruchinin_, Mar 16 2011
%t T[n_, k_] := (k/(2*n - k))*Sum[Binomial[2*n - k, n - k - j]*Binomial[2*n - k + j - 1, 2*n - k - 1], {j, 0, n - k}]; Table[T[n, k], {n, 0, 25}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Sep 05 2017 *)
%o (PARI) for(n=0,25, for(k=1,n, print1((k/(2*n-k))*sum(i=0,n-k, binomial(2*n-k,n-k-i)*binomial(2*n-k+i-1,2*n-k-1)), ", "))) \\ _G. C. Greubel_, Sep 05 2017
%Y Columns: A027307, A032349, A033296.
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Sep 15 2005
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