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%I #16 Dec 13 2023 08:53:22
%S 1,0,1,1,0,1,0,2,0,1,1,0,3,0,1,0,3,0,4,0,1,2,0,6,0,5,0,1,0,6,0,10,0,6,
%T 0,1,3,0,13,0,15,0,7,0,1,0,11,0,24,0,21,0,8,0,1,5,0,27,0,40,0,28,0,9,
%U 0,1,0,20,0,55,0,62,0,36,0,10,0,1
%N Triangle T(n, m)=Sum_{k=1..(n-m)/2} C(m+k-1, k)*T((n-m)/2, k), T(n,n)=1.
%F G.f.: A(x)^m=Sum_{n>=m} T(n,m)x^n, A(x)=Sum_{n>0} a(n)*x^(2*n-1), a(n) - is A000621.
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 0, 2, 0, 1;
%e 1, 0, 3, 0, 1;
%e 0, 3, 0, 4, 0, 1;
%e 2, 0, 6, 0, 5, 0, 1;
%p A253190 := proc(n,m)
%p option remember;
%p if n = m then
%p 1;
%p elif type(n-m,'odd') then
%p 0 ;
%p else
%p add(binomial(m+k-1,k)*procname((n-m)/2,k),k=1..(n-m)/2) ;
%p end if;
%p end proc: # _R. J. Mathar_, Dec 16 2015
%o (Maxima)
%o T(n, m):=if n=m then 1 else sum(binomial(m+k-1, k)*T((n-m)/2, k), k, 1, (n-m)/2);
%Y Cf. A000621 (row sums), A003600, A253184, A253189.
%K nonn,tabl
%O 1,8
%A _Vladimir Kruchinin_, Mar 24 2015
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