|
%I #12 Feb 26 2014 04:30:13
%S 1,2,-3,52,-1315,50286,-2655863,183322952,-15928677063,1695597280570,
%T -216636191518219,32688113040335292,-5749136647259226923,
%U 1165789270581830003942,-270019628802455686919295,70862777375461690495134736,-20921819854506620454336189583
%N E.g.f. satisfies: A(x) = x*exp( A(x)/A'(x) ).
%H Vaclav Kotesovec, <a href="/A193332/b193332.txt">Table of n, a(n) for n = 1..250</a>
%F E.g.f. A(x) satisfies:
%F (1) A(x)/A'(x) = -G(-x) where G(x) = x + x*G(x)*G'(x) is a g.f. of A088716; thus, log(A(x)/x) is an integer series.
%F (2) A(-L(-x)) = x where L(x) = x*exp(x*L'(x)) is the e.g.f. of A161968.
%F a(n) ~ c * (-1)^n * (n!)^2, where c = 0.217950789447151065... (see A238223). - _Vaclav Kotesovec_, Feb 26 2014
%e E.g.f.: A(x) = x + 2*x^2/2! - 3*x^3/3! + 52*x^4/4! - 1315*x^5/5! + 50286*x^6/6! - 2655863*x^7/7! + 183322952*x^8/8! +...
%e where A(x)/A'(x) = log(A(x)/x) equals the integer series:
%e (1) A(x)/A'(x) = x - x^2 + 3*x^3 - 14*x^4 + 85*x^5 - 621*x^6 + 5236*x^7 - 49680*x^8 + 521721*x^9 - 5994155*x^10 +...
%e which equals -G(-x) where G(x) is the g.f. of A088716.
%e The series reversion, -L(-x), begins:
%e (2) -L(-x) = x - 2*x^2/2! + 15*x^3/3! - 232*x^4/4! + 5905*x^5/5! - 220176*x^6/6! + 11210479*x^7/7! - 743759360*x^8/8! +...
%e where L(x) is the e.g.f. A161968.
%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(A/(A'+x*O(x^n))));n!*polcoeff(A,n)}
%Y Cf. A088716, A161968, A161967.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jul 23 2011
| 