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A253189 Triangle T(n, m)=Sum_{k=1..(n-m)/3} C(m, k)*T((n-m)/3, k), T(n,n)=1.

%I A253189 #17 Dec 13 2023 08:52:55
%S A253189 1,0,1,0,0,1,1,0,0,1,0,2,0,0,1,0,0,3,0,0,1,0,0,0,4,0,0,1,0,1,0,0,5,0,
%T A253189 0,1,0,0,3,0,0,6,0,0,1,0,0,0,6,0,0,7,0,0,1,0,0,0,0,10,0,0,8,0,0,1,0,0,
%U A253189 1,0,0,15,0,0,9,0,0,1,1,0,0,4,0,0,21,0,0,10,0,0,1,0,2,0,0,10,0,0,28,0,0,11,0,0,1,0,0,3,0,0,20,0,0,36,0,0,12,0,0,1
%N A253189 Triangle T(n, m)=Sum_{k=1..(n-m)/3} C(m, k)*T((n-m)/3, k), T(n,n)=1.
%H A253189 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%F A253189 G.f.: A(x)^m=Sum_{n>=m} T(n,m)*x^n, where A(x)=Sum_{n>0} x^((3^n-1)/2).
%e A253189 1;
%e A253189 0, 1;
%e A253189 0, 0, 1;
%e A253189 1, 0, 0, 1;
%e A253189 0, 2, 0, 0, 1;
%e A253189 0, 0, 3, 0, 0, 1;
%o A253189 (Maxima)
%o A253189 T(n, m):=if n=m then 1 else sum(binomial(m, k)*T((n-m)/3, k), k, 1, (n-m)/3);
%Y A253189 Cf. A003462, A113047, A253184, A253190.
%K A253189 nonn,tabl
%O A253189 1,12
%A A253189 _Vladimir Kruchinin_, Mar 24 2015