%I #6 May 08 2022 08:45:39

%S 1,1,3,39,983,36575,1820855,114756943,8825761159,806801625375,

%T 85941554164631,10503158853530607,1454442091007070247,

%U 225879929852989030463,39006369984778846202935,7435376490890285530440719,1554727341153441175044808647,354663953592562868624029236063

%N E.g.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * (exp(n*x) - A(x))^n / n!.

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

%C (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

%C here, q = exp(x) with p = -A(x), r = 2.

%F E.g.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} 2^n * (exp(n*x) - A(x))^n / n!.

%F (2) 1 = Sum_{n>=0} 2^n * exp(n^2*x - 2*A(x)*exp(n*x)) / n!.

%e E.g.f: A(x) = 1 + x + 3*x^2/2! + 39*x^3/3! + 983*x^4/4! + 36575*x^5/5! + 1820855*x^6/6! + 114756943*x^7/7! + 8825761159*x^8/8! + 806801625375*x^9/9! + ...

%e such that

%e 1 = 1 + 2*(exp(x) - A(x)) + 2^2*(exp(2*x) - A(x))^2/2! + 2^3*(exp(3*x) - A(x))^3/3! + 2^4*(exp(4*x) - A(x))^4/4! + 2^5*(exp(5*x) - A(x))^5/5! + 2^6*(exp(6*x) - A(x))^6/6! + ...

%e also

%e 1 = exp(-2*A(x)) + 2*exp(x - 2*A(x)*exp(x)) + 2^2*exp(4*x - 2*A(x)*exp(2*x))/2! + 2^3*exp(9*x - 2*A(x)*exp(3*x))/3! + 2^4*exp(16*x - 2*A(x)*exp(4*x))/4! + 2^5*exp(25*x - 2*A(x)*exp(5*x))/5! + ...

%e Related series.

%e log(A(x)) = x + 2*x^2/2! + 32*x^3/3! + 830*x^4/4! + 31384*x^5/5! + 1580150*x^6/6! + 100484768*x^7/7! + 7786595710*x^8/8! + 716464194216*x^9/9! + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

%o A[#A] = polcoeff( sum(m=0, #A, (exp(m*x +x*O(x^#A)) - Ser(A))^m * 2^m/m! ), #A-1)/2; ); H=A; n!*A[n+1]}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A326095, A353740.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 06 2022