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%I #16 Oct 22 2020 02:50:27
%S 1,1,3,23,323,7159,230277,10162541,590407925,43731367965,
%T 4025536525775,451018038948011,60450362738932847,9552672876260735595,
%U 1757890525033626257001,372703693123050652667001,90199912396771101321890057,24715956305580587510975488025,7612798125737548130715816004763
%N O.g.f. satisfies: A(x) = 1/(1 - x - x*( d/dx x^2*A(x)' )).
%C Compare to: F(x) = 1/(1 - x - x^2*F(x)'/F(x)) when F(x) = Sum_{n>=0} n!*x^n.
%H Vaclav Kotesovec, <a href="/A338301/b338301.txt">Table of n, a(n) for n = 0..250</a>
%F O.g.f. A(x) satisfies:
%F (1) A(x) = 1/(1 - x * d/dx (x + x^2*A(x)') ).
%F (2) 0 = [x^n] exp( n * (x + x^2*A(x)') ) / A(x), for n > 0.
%F a(n) ~ c * n!^2 / n, where c = 3.7361361487236058835975860887998... - _Vaclav Kotesovec_, Oct 22 2020
%e O.g.f.: A(x) = 1 + x + 3*x^2 + 23*x^3 + 323*x^4 + 7159*x^5 + 230277*x^6 + 10162541*x^7 + 590407925*x^8 + 43731367965*x^9 + 4025536525775*x^10 + ...
%e where
%e 1/A(x) = 1 - x - 2*x^2 - 18*x^3 - 276*x^4 - 6460*x^5 - 214770*x^6 - 9671634*x^7 - 569102296*x^8 - 42509370600*x^9 - ...
%e and
%e d/dx x^2*A(x)' = 2*x + 18*x^2 + 276*x^3 + 6460*x^4 + 214770*x^5 + 9671634*x^6 + 569102296*x^7 + 42509370600*x^8 + ...
%o (PARI) /* A(x) = 1/(1 - x - x*( d/dx x^2*A(x)' )) */
%o {a(n) = my(A=1+x); for(i=1, n, A=1/(1 - x - x*deriv(x^2*A'+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 31, print1(a(n), ", "))
%o (PARI) /* [x^n] exp( n * (x + x^2*A(x)') ) / A(x) = 0 */
%o {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(1+x*Ser(A)') ) / Ser(A) )[m] ); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A183607, A305110.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 21 2020
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