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A182969 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)).

%I A182969 #25 Jan 24 2025 18:27:40
%S A182969 1,1,4,23,159,1236,10454,94401,899286,8964253,92961432,998600238,
%T A182969 11075132605,126489183013,1484601117235,17876874457054,
%U A182969 220546820252773,2784446513061287,35940592329823310,473893641259375150
%N A182969 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)).
%F A182969 G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]*A(x)^(n+1)/(n+1)! ).
%F A182969 a(n) = T(n-1,1), where T(n,m) = (m/n)*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*k/i*binomial(2*i-k-1,i-1))*binomial(n+k-1,n-1)), n>m, T(n,n)=1. [_Vladimir Kruchinin_, May 07 2012]
%F A182969 T(n,m) = m * sum(k=1..n-m, (T(n-m,k)*binomial(n+2*k-1,n+k-1))/(n+k)) for n>m, and T(n,n) = 1. [_Vladimir Kruchinin_, Aug 08 2012]
%e A182969 G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 159*x^4 + 1236*x^5 +...
%e A182969 Related expansions:
%e A182969 A(x*A(x)) = 1 + x + 5*x^2 + 35*x^3 + 287*x^4 + 2592*x^5 + 25050*x^6 +...
%e A182969 A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 675*x^4 + 5331*x^5 + 45274*x^6 +...
%e A182969 Logarithmic series:
%e A182969 log(A(x)) = x*A(x)^2 + [d/dx x^3*A(x)^2]*A(x)^2/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^3/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^4/4! +...
%o A182969 (PARI) /* n-th Derivative: */
%o A182969 {Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
%o A182969 /* G.f.: */
%o A182969 {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
%o A182969 A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(m+1))*A^(m+1)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
%o A182969 (Maxima) T(n,m):=if n=m then 1 else m/n*sum(sum(T(n-m,i)*k/i*binomial(2*i-k-1,i-1),i,k,n-m)*binomial(n+k-1,n-1),k,1,n-m); makelist(T(n,1),n,1,10); /* _Vladimir Kruchinin_, May 07 2012 */
%Y A182969 Cf. A139702, A143426, A087949, A143435.
%K A182969 nonn
%O A182969 0,3
%A A182969 _Paul D. Hanna_, Dec 18 2010