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A212280 G.f. A(x)=1/(1-F(x)), where F(F(x)) = (1 - sqrt(1-16*x))/8.

%I A212280 #21 Apr 02 2014 19:20:10
%S A212280 1,1,3,17,131,1177,11531,119201,1276771,14015401,156585211,1772626673,
%T A212280 20275611347,233912585849,2718842818923,31816917837377,
%U A212280 374657837729987,4436890509548617
%N A212280 G.f. A(x)=1/(1-F(x)), where F(F(x)) = (1 - sqrt(1-16*x))/8.
%C A212280 F(x) is the generating function of A213422.
%H A212280 Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x)=F(x)</a>, arXiv:1302.1986
%F A212280 a(n) = sum(m=1..n, T(n,m)) for n>0, where T(n,m)= 1 if n=m, otherwise = (m *4^(n-m) *binomial(2*n-m-1,n-1)/n - sum_{i=m+1..n-1} T(n,i)*T(i,m) )/2.
%p A212280 T := proc(n,m)
%p A212280     if n = m then
%p A212280         1 ;
%p A212280     else
%p A212280         m*4^(n-m)*binomial(2*n-m-1,n-1)/n ;
%p A212280         %-add(procname(n,i)*procname(i,m),i=m+1..n-1) ;
%p A212280         %/2 ;
%p A212280     end if;
%p A212280 end proc:
%p A212280 A212280 := proc(n)
%p A212280     if n = 0 then
%p A212280         1
%p A212280     else
%p A212280         add(T(n,m),m=1..n) ;
%p A212280     end if;
%p A212280 end proc: # _R. J. Mathar_, Mar 04 2013
%t A212280 Clear[t]; t[n_, m_] := t[n, m] = 1/2*((m*4^(n-m)*Binomial[2*n-m-1, n-1]/n - Sum[ t[n, i]*t[i, m], {i, m+1, n-1}])); t[n_, n_] = 1; a[n_] := Sum[t[n, m], {m, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Feb 25 2013, from formula *)
%o A212280 (Maxima)
%o A212280 Solve(k):=block([Tmp,i,j],array(Tmp,k,k),for i:0 thru k do for j:0 thru k do Tmp[i,j]:a,
%o A212280 T(n,m):=if Tmp[n,m]=a then (if n=m then (Tmp[n,n]:1) else (Tmp[n,m]:(1/2*((m*4^(n-m)*binomial(2*n-m-1,n-1))/n-sum(T(n,i)*T(i,m),i,m+1,n-1))))) else Tmp[n,m],  makelist(sum(T(j,i),i,1,j),j,1,k));
%Y A212280 Cf. A213422.
%K A212280 nonn
%O A212280 0,3
%A A212280 _Vladimir Kruchinin_, Feb 14 2013