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A182954 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^4.

%I A182954 #12 Jun 05 2025 01:53:51
%S A182954 1,1,5,39,381,4284,53163,710810,10085621,150326044,2336828792,
%T A182954 37687170215,628069684439,10782885724300,190248852445782,
%U A182954 3442896376032300,63804661588968521,1209314277690837796
%N A182954 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x) )^4.
%H A182954 Paul D. Hanna, <a href="/A182954/b182954.txt">Table of n, a(n) for n = 0..300</a>
%F A182954 G.f. A(x) satisfies:
%F A182954 * A(x) = exp( Sum_{m>=0} {d^m/dx^m x^m*A(x)^(4m+4)} * x^(m+1)/(m+1)! );
%F A182954 * A(x) = exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^(4m)}*x^k]*x^m/m);
%F A182954 which are equivalent.
%F A182954 Recurrence:
%F A182954 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
%F A182954 a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,4k).
%e A182954 G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + ...
%e A182954 Related expansions:
%e A182954 A(x*A(x)) = 1 + x + 6*x^2 + 54*x^3 + 592*x^4 + 7331*x^5 + 98870*x^6 + ...
%e A182954 A(x*A(x))^4 = 1 + 4*x + 30*x^2 + 292*x^3 + 3305*x^4 + 41420*x^5 + ...
%e A182954 The g.f. satisfies:
%e A182954 log(A(x)) = A(x)^4*x + {d/dx x*A(x)^8}*x^2/2! + {d^2/dx^2 x^2*A(x)^12}*x^3/3! + {d^3/dx^3 x^3*A(x)^16}*x^4/4! + ...
%o A182954 (PARI) {a(n) = my(A = 1 + sum(i=1,n-1,a(i)*x^i+x*O(x^n)));
%o A182954 for(i=1,n, A = exp( sum(m=1,n, sum(k=0,n-m, binomial(m+k-1,k)*polcoef(A^(4*m),k)*x^k) * x^m/m ) + x*O(x^n))); polcoef(A,n)}
%o A182954 (PARI) {a(n, m=1) = if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, 4*k))))}
%Y A182954 Cf. A384265, A030266, A121687, A182953, A182955.
%K A182954 nonn
%O A182954 0,3
%A A182954 _Paul D. Hanna_, Dec 15 2010