|
%I #4 Jun 06 2019 18:25:12
%S 1,1,2,15,234,5525,176823,7232050,363749900,21891574683,1544392825386,
%T 125684334518985,11648104664937271,1216426938053726672,
%U 141882106115149781072,18344653087551340567427,2612237303669636927142962,407360290222179197806752141,69221669418346150774013957483,12760799611726977737400430776570,2541919524892729009158279375599352
%N E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (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 = 1.
%H Paul D. Hanna, <a href="/A326095/b326095.txt">Table of n, a(n) for n = 0..195</a>
%F E.g.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} (exp(n*x) - A(x))^n / n!.
%F (2) 1 = Sum_{n>=0} exp(n^2*x - A(x)*exp(n*x)) / n!.
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 234*x^4/4! + 5525*x^5/5! + 176823*x^6/6! + 7232050*x^7/7! + 363749900*x^8/8! + 21891574683*x^9/9! + 1544392825386*x^10/10! + ...
%e such that
%e 1 = 1 + (exp(x) - A(x)) + (exp(2*x) - A(x))^2/2! + (exp(3*x) - A(x))^3/3! + (exp(4*x) - A(x))^4/4! + (exp(5*x) - A(x))^5/5! + (exp(6*x) - A(x))^6/6! + ...
%e also
%e 1 = exp(-A(x)) + exp(x - A(x)*exp(x)) + exp(4*x - A(x)*exp(2*x))/2! + exp(9*x - A(x)*exp(3*x))/3! + exp(16*x - A(x)*exp(4*x))/4! + exp(25*x - A(x)*exp(5*x))/5! + exp(36*x - A(x)*exp(6*x))/6! + ...
%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/m! ),#A-1);); n!*A[n+1]}
%o for(n=0,25,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 06 2019
| 