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A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

%I A173895 #18 Nov 02 2019 03:13:43
%S A173895 1,1,-1,0,9,-48,15,2448,-24927,23424,3091311,-47659200,88056969,
%T A173895 10702667520,-225139993377,679791291648,78646340795265,
%U A173895 -2128005345251328,9456106738649631,1053535684549174272
%N A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.
%C A173895 Define a polynomial sequence P_n(x) recursively by
%C A173895 ... P_0(x) = 1, and for n >= 1
%C A173895 ... P_n(x) = (x-1)*P_(n-1)(x-1)-n*P_(n-1)(x+1).
%C A173895 The first few polynomials are
%C A173895 P_1(x) = x-2
%C A173895 P_2(x) = x^2-6*x+5
%C A173895 P_3(x) = x^3-12*x^2+32*x-12.
%C A173895 It appears that a(n+1) = P_n(1) (checked as far as a(19)).
%C A173895 Compare with A144010.
%H A173895 Vaclav Kotesovec, <a href="/A173895/b173895.txt">Table of n, a(n) for n = 0..190</a>
%F A173895 E.g.f. satisfies: A(x) = 1 + Integral 1/(1 + x*A(x)) dx.
%F A173895 E.g.f. satisfies: A(G(x)) = 1 + x where G(x) is the e.g.f. of A000932 (offset 1). [_Paul D. Hanna_, Aug 23 2011]
%e A173895 E.g.f.: A(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! +...
%e A173895 where
%e A173895 1/(1 + x*A(x)) = 1 - x + 9*x^3/3! - 48*x^4/4! + 15*x^5/5! + 2448*x^6/6! +...
%e A173895 Also, A(G(x)) = 1 + x where
%e A173895 G(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + 492*x^8/8! +...+ A000932(n-1)*x^n/n! +...
%t A173895 m = 20; A[_] = 1;
%t A173895 Do[A[x_] = 1 + Integrate[1/(1+x*A[x])+O[x]^m, x]+O[x]^m // Normal, {m}];
%t A173895 CoefficientList[A[x], x] * Range[0, m-1]! (* _Jean-François Alcover_, Nov 02 2019 *)
%o A173895 (PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1+x*A+x*O(x^n)) ));n!*polcoeff(A, n)}
%Y A173895 Cf. A144010, A000932.
%K A173895 easy,sign
%O A173895 0,5
%A A173895 _Peter Bala_, Nov 26 2010