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

%I #21 Apr 02 2014 19:20:10

%S 1,1,3,17,131,1177,11531,119201,1276771,14015401,156585211,1772626673,

%T 20275611347,233912585849,2718842818923,31816917837377,

%U 374657837729987,4436890509548617

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

%C F(x) is the generating function of A213422.

%H 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 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 T := proc(n,m)

%p if n = m then

%p 1 ;

%p else

%p m*4^(n-m)*binomial(2*n-m-1,n-1)/n ;

%p %-add(procname(n,i)*procname(i,m),i=m+1..n-1) ;

%p %/2 ;

%p end if;

%p end proc:

%p A212280 := proc(n)

%p if n = 0 then

%p 1

%p else

%p add(T(n,m),m=1..n) ;

%p end if;

%p end proc: # _R. J. Mathar_, Mar 04 2013

%t 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 (Maxima)

%o 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 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 Cf. A213422.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Feb 14 2013