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%I #6 Oct 06 2020 05:07:25
%S 1,1,6,54,628,8760,140904,2552151,51243864,1127982321,26993774100,
%T 697703846499,19372450060296,575205186725962,18191422973198622,
%U 610655961723782310,21689599103526363600,812832263931582168447,32057155649057309677062,1327393477257351399000744,57581802198755959140129600
%N G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].
%H Vaclav Kotesovec, <a href="/A305602/b305602.txt">Table of n, a(n) for n = 0..400</a>
%F O.g.f. A(x) satisfies:
%F (1) [x^n] exp( n * Integral A(x)^2 dx ) * (n + 1 - A(x)) = 0 for n > 0.
%F (2) A(x) = 1 + x*A(x)*(A(x) + 2*x*A'(x))/(1 - x*A(x)^2)^2.
%F a(n) ~ c * 2^n * n^(3/2) * n!, where c = 0.26934871195193907483980578... - _Vaclav Kotesovec_, Oct 06 2020
%e G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 628*x^4 + 8760*x^5 + 140904*x^6 + 2552151*x^7 + 51243864*x^8 + 1127982321*x^9 + 26993774100*x^10 + ...
%e such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 13*x^2 + 120*x^3 + 1400*x^4 + 19424*x^5 + 309780*x^6 + 5559054*x^7 + 110623342*x^8 + 2415298374*x^9 + 57387784542*x^10 + ...
%e 1/(1 - x*A(x)^2) = 1 + x + 3*x^2 + 18*x^3 + 157*x^4 + 1752*x^5 + 23484*x^6 + 364593*x^7 + 6405483*x^8 + 125331369*x^9 + 2699377410*x^10 + ...
%e exp( Integral A(x)^2 dx ) = 1 + x + 3*x^2/2! + 33*x^3/3! + 849*x^4/4! + 38061*x^5/5! + 2575611*x^6/6! + 242377533*x^7/7! + 30085188993*x^8/8! + ...
%e A'(x)/A(x) = 1 + 11*x + 145*x^2 + 2247*x^3 + 39461*x^4 + 768983*x^5 + 16409646*x^6 + 380013063*x^7 + 9487631035*x^8 + 254076973011*x^9 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^2+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^2) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A305603, A305604, A209881, A305110.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 05 2018
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