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%I #30 Apr 25 2024 08:25:42
%S 1,1,2,7,34,210,1574,13866,140340,1604284,20439484,287152488,
%T 4409695952,73482586464,1320533540808,25456195929232,523975944225280,
%U 11469534961767408,266038450202037728,6518167274358688512,168209881653024622944,4560447490191133853536,129593625015740116555072
%N E.g.f. satisfies: A(x) = 1 + Integral A(x) + A(x)^2*log(A(x)) dx.
%C Compare to e.g.f. of Bell numbers: if B(x) = exp(exp(x)-1) then
%C B(x) = 1 + Integral B(x) + B(x)*log(B(x)) dx.
%C Limit_{n->oo} (a(n)/n!)^(1/n) = 1.30339... (cf. A235129). - _Vaclav Kotesovec_, Nov 09 2014
%H Vaclav Kotesovec, <a href="/A249833/b249833.txt">Table of n, a(n) for n = 0..230</a>
%F E.g.f. satisfies: A(x) = exp( Integral 1 + A(x)*log(A(x)) dx ).
%F Conjecture: a(n) = A370382(n-1, 0) for n > 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 25 2024
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 210*x^5/5! + ...
%e Related expansions.
%e A(x)^2*log(A(x)) = x + 5*x^2/2! + 27*x^3/3! + 176*x^4/4! + 1364*x^5/5! + ...
%e A(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 148*x^4/4! + 1040*x^5/5! + 8688*x^6/6! + 84068*x^7/7! + 924384*x^8/8! + 11381696*x^9/9! + ...
%e log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 64*x^5/5! + 424*x^6/6! + 3358*x^7/7! + 30952*x^8/8! + 325488*x^9/9! + 3845724*x^10/10! + ...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+intformal(A+A^2*log(A +x*O(x^n))));n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(1+A*log(A +x*O(x^n)))));n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A235129.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 06 2014
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