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A386645 E.g.f. A(x) satisfies A(x) = Sum_{n>=0} ( A(x)^n + log(A(x)) )^n * x^n / n!.

%I A386645 #14 Aug 11 2025 10:31:22
%S A386645 1,1,5,46,665,13416,350227,11254300,430093617,19067201056,
%T A386645 962456078051,54518610032844,3425698051345561,236531783119320352,
%U A386645 17805560350371525747,1451679746048430507676,127461102015439274388833,11993733613161390301999680,1204348142606115777871016899,128572080570190461521783550988
%N A386645 E.g.f. A(x) satisfies A(x) = Sum_{n>=0} ( A(x)^n + log(A(x)) )^n * x^n / n!.
%C A386645 It appears that for n > 6, a(n) (mod 6) equals [1, 4, 3, 4, 5, 0] repeating.
%C A386645 In general, the following sums are equal:
%C A386645 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386645 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386645 here, q = A(x) with p = log(A(x)), r = x.
%H A386645 Paul D. Hanna, <a href="/A386645/b386645.txt">Table of n, a(n) for n = 0..230</a>
%F A386645 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386645 (1) A(x) = Sum_{n>=0} A(x)^(n^2) * A(x)^(x*A(x)^n) * x^n / n!.
%F A386645 (2) A(x) = Sum_{n>=0} ( A(x)^n + log(A(x)) )^n * x^n / n!.
%e A386645 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 46*x^3/3! + 665*x^4/4! + 13416*x^5/5! + 350227*x^6/6! + 11254300*x^7/7! + 430093617*x^8/8! + 19067201056*x^9/9! + ...
%e A386645 where
%e A386645 A(x) = A(x)^x + A(x)*A(x)^(x*A(x))*x + A(x)^4*A(x)^(x*A(x)^2)*x^2/2! + A(x)^9*A(x)^(x*A(x)^3)*x^3/3! + A(x)^16*A(x)^(x*A(x)^4)*x^4/4! + ...
%e A386645 Also,
%e A386645 A(x) = 1 + (A(x) + log(A(x)))*x + (A(x)^2 + log(A(x)))^2*x^2/2! + (A(x)^3 + log(A(x)))^3*x^3/3! + (A(x)^4 + log(A(x)))^4*x^4/4! + ...
%e A386645 RELATED SERIES.
%e A386645 log(A(x)) = x + 4*x^2/2! + 33*x^3/3! + 460*x^4/4! + 9185*x^5/5! + 239406*x^6/6! + 7704403*x^7/7! + 295172872*x^8/8! + 13123492929*x^9/9! + 664403960890*x^10/10! + ...
%o A386645 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^m + log(Ser(A)))^m * x^m/m! ), #A-1) ); n!*A[n+1]}
%o A386645 for(n=0, 20, print1(a(n), ", "))
%Y A386645 Cf. A386644.
%K A386645 nonn
%O A386645 0,3
%A A386645 _Paul D. Hanna_, Aug 08 2025