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%I #5 Nov 21 2017 05:28:07
%S 1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,298370920,
%T -4372151566,68234087624,-1129894265272,19788479904366,
%U -365520041466291,7103187300763530,-144897616964143050,3096285550330959336
%N G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].
%H Vaclav Kotesovec, <a href="/A158884/b158884.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
%F G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
%F G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
%F G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
%F G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
%F a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - _Vaclav Kotesovec_, Nov 21 2017
%e G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
%e d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
%e d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
%e Coefficients in powers A(x)^-n begin:
%e A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
%e A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
%e A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
%e A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
%e A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
%e A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
%e A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
%e A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
%e where coefficients in parenthesis form A158883 and signed A088716
%e and A(x)^-1 (first row) is the g.f. of signed A088715.
%o (PARI) {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);Vec(Ser(A)^(n+1)/(n+1))[n+1]}
%Y Cf. A158883, A088715, A088716.
%K sign
%O 0,4
%A _Paul D. Hanna_, Apr 30 2009
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