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A351917 G.f. A(x) = lim_{n->infinity} F(n), where F(0) = 1, F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1,2,3,...

%I A351917 #12 Feb 27 2022 03:58:49
%S A351917 1,1,2,7,46,428,5297,79803,1425901,29317826,682661738,17737351760,
%T A351917 508899803500,15976473483553,544826377810898,20054787151757636,
%U A351917 792552513383685078,33469556711571645662,1504188914267750552104,71681627511482293102424,3610465782547332984625361
%N A351917 G.f. A(x) = lim_{n->infinity} F(n), where F(0) = 1, F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1,2,3,...
%H A351917 Vaclav Kotesovec, <a href="/A351917/b351917.txt">Table of n, a(n) for n = 0..380</a>
%F A351917 a(n) ~ c * n^(n-1), where c = A072364 = exp(-exp(-1)). - _Vaclav Kotesovec_, Feb 27 2022
%e A351917 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + 1425901*x^8 + 29317826*x^9 + 682661738*x^10 + ...
%e A351917 The g.f. A(x) equals the limit of the following process.
%e A351917 Start with F(0) = 1, then repeat F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1..N, for some positive integer N; the g.f. of this sequence equals the limit as N approaches infinity.
%e A351917 Example: start with F(0) = 1, then continue as follows.
%e A351917 F(1) = ( F(0)^1 + (1*x)^1 )^(1/1) = 1 + x;
%e A351917 F(2) = ( F(1)^2 + (2*x)^2 )^(1/2) = 1 + x + 2*x^2 - 2*x^3 + 4*x^5 - 6*x^6 - 2*x^7 + 22*x^8 + ...;
%e A351917 F(3) = ( F(2)^3 + (3*x)^3 )^(1/3) = 1 + x + 2*x^2 + 7*x^3 - 18*x^4 - 5*x^5 + 21*x^6 + 232*x^7 + ...;
%e A351917 F(4) = ( F(3)^4 + (4*x)^4 )^(1/4) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 - 197*x^5 + 21*x^6 - 216*x^7 + ...;
%e A351917 F(5) = ( F(4)^5 + (5*x)^5 )^(1/5) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 - 2479*x^6 + 1034*x^7 + ...;
%e A351917 F(6) = ( F(5)^6 + (6*x)^6 )^(1/6) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 - 37846*x^7 + ...;
%e A351917 F(7) = ( F(6)^7 + (7*x)^7 )^(1/7) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + ...;
%e A351917 ...
%e A351917 Continue in this way to obtain the g.f. A(x) as a limit of F(n) as n approaches infinity.
%e A351917 Related table.
%e A351917 The table of coefficients of x^k in A(x)^n, for k >= 0 and n >= 1, begins:
%e A351917 n=1: [1, 1,  2,   7,   46,  428,  5297,   79803, ...];
%e A351917 n=2: [1, 2,  5,  18,  110,  976, 11683,  172556, ...];
%e A351917 n=3: [1, 3,  9,  34,  198, 1677, 19388,  280326, ...];
%e A351917 n=4: [1, 4, 14,  56,  317, 2572, 28694,  405564, ...];
%e A351917 n=5: [1, 5, 20,  85,  475, 3711, 39945,  551180, ...];
%e A351917 n=6: [1, 6, 27, 122,  681, 5154, 53558,  720630, ...];
%e A351917 n=7: [1, 7, 35, 168,  945, 6972, 70035,  918016, ...]; ...
%e A351917 in which row n is comparable with F(n)^n given in the generating process shown above.
%e A351917 Related series.
%e A351917 log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 155*x^4/4 + 1886*x^5/5 + 28908*x^6/6 + 517539*x^7/7 + 10708979*x^8/8 + 250010863*x^9/9 + 6513735398*x^10/10 + ...
%o A351917 (PARI) {a(n) = my(A = 1 +x*O(x^n)) ; for(m=1,n, A = (A^m + (m*x)^m )^(1/m) ); polcoeff(A,n)}
%o A351917 for(n=0,30,print1(a(n),", "))
%K A351917 nonn
%O A351917 0,3
%A A351917 _Paul D. Hanna_, Feb 25 2022