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%I #26 Feb 09 2018 03:10:18
%S 1,1,1,4,21,160,1525,17760,243145,3833600,68373225,1361264000,
%T 29925477725,719991897600,18817847565725,530921477363200,
%U 16082605690148625,520603130117939200,17934634668874889425
%N E.g.f. satisfies: A'(x) = 1/(1 - x*A(x)) with A(0)=1.
%C From _Peter Bala_, Nov 26 2010: (Start)
%C Define a polynomial sequence P_n(x) recursively by
%C ... P_0(x) = 1,
%C ... P_n(x) = (x-1)*P_(n-1)(x-1) + n*P_(n-1)(x+1) for n >= 1.
%C The first few polynomials are
%C P_1(x) = x;
%C P_2(x) = x^2 + 3;
%C P_3(x) = x^3 + 12*x + 8.
%C It appears that a(n+1) = P_n(1) (checked as far as a(19)).
%C Compare with A173895. (End)
%H Vaclav Kotesovec, <a href="/A144010/b144010.txt">Table of n, a(n) for n = 0..400</a>
%F E.g.f. satisfies: A(x) = 1 + Integral 1/(1 - x*A(x)) dx.
%F a(n) ~ n^(n-1) * s^n / exp(n), where s = 2.0832144900084392272885741721727173082215... is the root of the equation sqrt(Pi/2)*s*exp(-s^2/2)*(erfi(1/sqrt(2)) - erfi(s/sqrt(2))) = -1. - _Vaclav Kotesovec_, Feb 23 2014
%F a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * binomial(n-1, k) * a(k) * a(n-k-1). - _Vladimir Reshetnikov_, May 17 2016
%t FindRoot[Sqrt[Pi/2]*s*E^(-s^2/2)*(Erfi[1/Sqrt[2]]-Erfi[s/Sqrt[2]]) == -1,{s,1},WorkingPrecision->50] (* program for numerical value of the constant s, _Vaclav Kotesovec_, Feb 23 2014 *)
%t a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k Binomial[n-1, k] a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, May 17 2016 *)
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* from _Peter Bala_'s Formula */
%o {a(n)=local(P=1);if(n>=0&n<2,1,for(k=1,n-1,P=(x-1)*subst(P,x,x-1) + k*subst(P,x,x+1)));subst(P,x,1)}
%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Jun 15 2013
%Y Cf. A144011, A238302.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 10 2008
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