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%I #26 Mar 20 2018 10:31:22
%S 1,1,4,24,184,1672,17296,198800,2499200,33992000,496281344,7731823616,
%T 127946465280,2240485196800,41387447564800,804353715776000,
%U 16408115358117888,350584123058300928,7831051680901885952,182550106828365115392,4433782438058087202816,112031844502468602085376,2940834866411162315849728
%N O.g.f. A(x) satisfies: A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).
%C O.g.f. equals the logarithm of the e.g.f. of A300735.
%C The e.g.f. G(x) of A300735 satisfies: [x^n] G(x)^(2*n) = (n+1) * [x^(n-1)] G(x)^(2*n) for n>=1.
%H Paul D. Hanna, <a href="/A300736/b300736.txt">Table of n, a(n) for n = 1..200</a>
%F O.g.f. A(x) satisfies: [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) for n>=1.
%F a(n) ~ c * n! * n^3, where c = 0.0087891365985... - _Vaclav Kotesovec_, Mar 20 2018
%e O.g.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 184*x^5 + 1672*x^6 + 17296*x^7 + 198800*x^8 + 2499200*x^9 + 33992000*x^10 + 496281344*x^11 + 7731823616*x^12 + ...
%e where
%e A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 697*x^4/4! + 25761*x^5/5! + 1371691*x^6/6! + 97677343*x^7/7! + 8869533681*x^8/8! + 993709302337*x^9/9! + 134086553693011*x^10/10! + ... + A300735(n)*x^n/n! + ...
%e A'(x) = 1 + 2*x + 12*x^2 + 96*x^3 + 920*x^4 + 10032*x^5 + 121072*x^6 + 1590400*x^7 + 22492800*x^8 + 339920000*x^9 + 5459094784*x^10 + ...
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = x*(1-x*A')/(1-2*x*A' +x*O(x^n))); polcoeff(A,n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) /* [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) */
%o {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(2*(#A-1))); A[#A] = ((#A)*V[#A-1] - V[#A])/(2*(#A-1)) ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A088716, A300987, A300989, A300991, A300993.
%Y Cf. A300735, A088716, A300591, A296171, A300593, A300595.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 17 2018
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