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%I #24 Apr 12 2024 04:27:45
%S 1,-1,7,-37,371,-4741,72885,-1380541,29678815,-726074821,19834534193,
%T -597434105005,19716033256947,-706675332962509,27351721308658141,
%U -1136955116183424829,50513209770352997927,-2388911790071698253845,119817073596530701766985,-6352554087532686682163053
%N Expansion of e.g.f. A(x) satisfying Sum_{n>=0} (A(x)^n - x)^n / n! = 1.
%C Related identity: Sum_{n>=0} (q^n + p)^n * r^n/n! = Sum_{n>=0} exp(p*q^n*r) * q^(n^2) * r^n/n!; here, q = A(x), p = -x, r = 1.
%C Conjecture: for n > 0, a(6*n + k) == [2,0,2,1,2,2] (mod 3) at k = [0,1,2,3,4,5], respectively.
%H Paul D. Hanna, <a href="/A371710/b371710.txt">Table of n, a(n) for n = 1..300</a>
%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F (1) Sum_{n>=0} (A(x)^n - x)^n / n! = 1.
%F (2) Sum_{n>=0} exp(-x*A(x)^n) * A(x)^(n^2) / n! = 1.
%F a(n) ~ -c * n^(n-1) / (exp(n) * r^n), where c = sqrt( (-r*Sum_{k>=0} ((k*(-r + s^k)^(-1 + k))/k!)) / Sum_{k>=0} ((-1 + k)*k^2 * s^(-2 + k)*(-r + s^k)^(-2 + k)*(-r + (1 + k)*s^k))/k! ) = 0.4187940561612508319941857365856199965995537726..., and where r = -0.3491010753747466229482161022113556770139942642631... and s = -0.6426571454882319173136663034041471668334385049965... are roots of the equations Sum_{k>=0} (-r + s^k)^k/k! = 1 and Sum_{k>=0} (k^2*s^(-1 + k)*(-r + s^k)^(-1 + k))/k! = 0. - _Vaclav Kotesovec_, Apr 11 2024
%F The values r and A(r) = s given above also satisfy Sum_{n>=0} n*(n - r*A(r)^n) * exp(-r*A(r)^n) * A(r)^(n^2) / n! = 0. - _Paul D. Hanna_, Apr 12 2024
%e E.g.f.: A(x) = x - x^2/2! + 7*x^3/3! - 37*x^4/4! + 371*x^5/5! - 4741*x^6/6! + 72885*x^7/7! - 1380541*x^8/8! + 29678815*x^9/9! - 726074821*x^10/10! + ...
%e where e.g.f. A(x) satisfies the following sums.
%e (1) 1 = 1 + (A(x) - x) + (A(x)^2 - x)^2/2! + (A(x)^3 - x)^3/3! + (A(x)^4 - x)^4/4! + (A(x)^5 - x)^5/5! + ...
%e (2) 1 = exp(-x) + exp(-x*A(x))*A(x) + exp(-x*A(x)^2)*A(x)^4/2! + exp(-x*A(x)^3)*A(x)^9/3! + exp(-x*A(x)^4)*A(x)^16/4! + ...
%e SPECIFIC VALUES.
%e A(1/3) = 0.309336999832107073180903710282149168034207161078640395207...
%e A(1/4) = 0.232922937634173409470673241764259081533730452334005659588...
%e A(1/5) = 0.187560199855301209894398645611115284037479048219241021351...
%e A(-1/3) = -0.5146620783815103062311605400508155869729182062358910349...
%e A(-1/4) = -0.3123628005245983090140211998639545568283470783996606926...
%e A(-1/5) = -0.2335665203884038676850050992335539648367581317265287642...
%o (PARI) /* Sum_{n>=0} exp(-x*A(x)^n) * A(x)^(n^2) / n! = 1 */
%o {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A[#A] = -polcoeff( sum(m=0,sqrtint(#A+1), exp(-x*Ser(A)^m +x*O(x^#A)) * Ser(A)^(m^2)/m! ), #A-1); ); n!*A[n+1]}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) /* Sum_{n>=0} (A(x)^n - x)^n / n! = 1 */
%o {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A[#A] = -polcoeff( sum(m=0,#A+1, (Ser(A)^m - x)^m/m! ), #A-1) ); n!*A[n+1]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A326600, A326097, A326433.
%K sign
%O 1,3
%A _Paul D. Hanna_, Apr 10 2024
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