This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369091 #11 Jan 29 2024 10:23:49
%S A369091 1,2,6,36,260,2190,21882,268856,3907080,63977850,1152946190,
%T A369091 22581979332,477140664156,10828556474918,263163922847490,
%U A369091 6836792356168560,189694001088036752,5614994984290505586,176964200467784915094,5921022573291003915260,209568707084236321665060
%N A369091 Expansion of e.g.f. A(x) satisfying A(x) = x + A( x^2*exp(x) ), with A(0) = 0.
%C A369091 Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).
%H A369091 Paul D. Hanna, <a href="/A369091/b369091.txt">Table of n, a(n) for n = 1..300</a>
%F A369091 E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
%F A369091 (1) A(x) = x + A( x^2*exp(x) ).
%F A369091 (2) A(x) = Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
%F A369091 (3) A(x) = log(G(x)/x) where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
%e A369091 E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + 3907080*x^9/9! + 63977850*x^10/10! + ...
%e A369091 which equals the sum of all iterations of the function x^2*exp(x).
%e A369091 RELATED SERIES.
%e A369091 x*exp(A(x)) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! + 4206*x^6/6! + 48307*x^7/7! + 632360*x^8/8! + ... + A369090(n)*x^n/n! + ...
%e A369091 Let R(x) be the series reversion of A(x),
%e A369091 R(x) = x - 2*x^2/2! + 6*x^3/3! - 36*x^4/4! + 340*x^5/5! - 3870*x^6/6! + 52038*x^7/7! - 850472*x^8/8! + 16378920*x^9/9! + ...
%e A369091 then R(x) and e.g.f. A(x) satisfy:
%e A369091 (1) R( A(x) ) = x,
%e A369091 (2) R( A(x) - x ) = x^2 * exp(x).
%e A369091 GENERATING METHOD.
%e A369091 Let F(n) equal the n-th iteration of x^2*exp(x), so that
%e A369091 F(0) = x,
%e A369091 F(1) = x^2 * exp(x),
%e A369091 F(2) = x^4 * exp(2*x) * exp(x^2*exp(x)),
%e A369091 F(3) = x^8 * exp(4*x) * exp(2*x^2*exp(x)) * exp(F(2)),
%e A369091 F(4) = x^16 * exp(8*x) * exp(4*x^2*exp(x)) * exp(2*F(2)) * exp(F(3)),
%e A369091 F(5) = x^32 * exp(16*x) * exp(8*x^2*exp(x)) * exp(4*F(2)) * exp(2*F(3)) * exp(F(4)),
%e A369091 ...
%e A369091 F(n+1) = F(n)^2 * exp(F(n))
%e A369091 ...
%e A369091 Then the e.g.f. A(x) equals the sum
%e A369091 A(x) = F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...
%e A369091 equivalently,
%e A369091 A(x) = x + x^2*exp(x) + x^4*exp(2*x)*exp(x^2*exp(x)) + x^8*exp(4*x)*exp(2*x^2*exp(x)) * exp(x^4*exp(2*x)*exp(x^2*exp(x))) + ...
%o A369091 (PARI) {a(n) = my(A=x); for(i=0, #binary(n),
%o A369091 A = x + subst(A, x, x^2*exp(x +x^2*O(x^n)) )); n! * polcoeff(A, n)}
%o A369091 for(n=1, 30, print1(a(n), ", "))
%Y A369091 Cf. A369090, A369551 (a(n)/n), A030178.
%K A369091 nonn
%O A369091 1,2
%A A369091 _Paul D. Hanna_, Jan 26 2024