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%I #32 Jun 03 2019 12:27:11
%S 1,1,1,3,2,1,10,7,3,1,38,26,12,4,1,154,105,49,18,5,1,654,444,210,80,
%T 25,6,1,2871,1944,927,363,120,33,7,1,12925,8734,4191,1672,575,170,42,
%U 8,1,59345,40040,19305,7810,2761,858,231,52,9,1
%N Triangle T(n,m) read by rows, [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) satisfies A(x) = x/(1-A(x)-A(x)^2).
%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(n,m) = A037027(2*n-m-1,n-1)*m/n.
%F T(n,m) = Sum_{i=1..n-m+1} A001002(i-1)*T(n-i,m-1), m>0. T(n,1) = A001002(n-1).
%F T(n,m) = if n<0 or m<0 or n<m then 0 else if n=m then 1 else if m=0 then 0 else T(n-1,m-1)+T(n,m+1)+T(n,m+2). - _Vladimir Kruchinin_, Apr 21 2016
%F Assuming the range n>=0 and 0<=k<=n we have T(n,k) = C(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1)) for n>=1 and T(0,0) = 1. - _Peter Luschny_, Apr 25 2016
%e Triangle starts:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 10, 7, 3, 1;
%e 38, 26, 12, 4, 1;
%e 154, 105, 49, 18, 5, 1;
%e ...
%p T := (n,k) -> `if`(n=0,1,binomial(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1)): seq(seq(simplify(T(n,k), k=0..n)), n=0..10); # _Peter Luschny_, Apr 25 2016
%t T[n_, m_] := T[n, m] = Which[n <= 0 || m <= 0, 0, n < m, 0, n == m, 1, True, T[n-1, m-1] + T[n, m+1] + T[n, m+2]];
%t Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 03 2019, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o T(n, m):=if n<=0 or m<=0 then 0 else if n<m then 0 else if n=m then 1 else T(n-1, m-1)+T(n, m+1)+T(n, m+2); /* _Vladimir Kruchinin_, Apr 21 2016 */
%Y Cf. A001002, A037027.
%K nonn,tabl
%O 1,4
%A _Vladimir Kruchinin_, Mar 21 2011
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