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G.f. A(x) satisfies A(A(x)) = x + A(2*x)^2.
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%I #17 Feb 10 2025 09:32:34

%S 1,2,12,160,4592,276496,34174592,8570174016,4335215019520,

%T 4408454839564672,8992935435667848448,36753720073439398166016,

%U 300717909357395506394597376,4923649248081508021291300507648,161279005396471404160636537186082816,10567483324648990167101852566698802679808

%N G.f. A(x) satisfies A(A(x)) = x + A(2*x)^2.

%F a(n)=T(n,1), T(n,m) = (1/2)*(kron_delta(n,m) + (Sum_{j=max(0,2*m-n)..m-1} binomial(m,j)*2^(n-j)*T(n-j,2*(m-j))) - (Sum_{k=m+1..n-1} T(n,k)*T(k,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, May 03 2012

%e G.f.: A(x) = x + 2*x^2 + 12*x^3 + 160*x^4 + 4592*x^5 + 276496*x^6 +...

%e A(A(x)) = x + 4*x^2 + 32*x^3 + 448*x^4 + 11776*x^5 + 637952*x^6 +...

%e A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 368*x^5 + 9968*x^6 + 575200*x^7 +...

%o (PARI) {a(n)=local(A=x+sum(k=2,n-1,a(k)*x^k)+x*O(x^n));if(n==1,1,polcoeff(x+subst(A,x,2*x)^2-subst(A,x,A),n)/2)}

%o (Maxima) T(n,m):=( if n=m then 1 else 1/2*(kron_delta(n,m)+ sum(binomial(m,j)*2^(n-j)*T(n-j,2*(m-j)),j,max(0,2*m-n),m-1)-sum(T(n,k)*T(k,m),k,m+1,n-1))); makelist(T(n,1),n,1,7); /* _Vladimir Kruchinin_, May 03 2012 */

%K nonn,changed

%O 1,2

%A _Paul D. Hanna_, Sep 03 2010

%E More terms from _Michel Marcus_, Feb 10 2025