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%I #16 Mar 20 2018 10:33:23
%S 1,1,8,84,1080,16056,266256,4816080,93638016,1937252160,42339628800,
%T 972303685632,23365476089856,585706819083264,15276194983411200,
%U 413695882240574976,11612673418376392704,337392794531354462208,10133165365696293507072,314252173854006410465280,10053170842576476899524608,331455812860465669006442496
%N O.g.f. A(x) satisfies: A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).
%C O.g.f. equals the logarithm of the e.g.f. of A300992.
%C The e.g.f. G(x) of A300992 satisfies: [x^n] G(x)^(6*n) = (n+5) * [x^(n-1)] G(x)^(6*n) for n>=1.
%H Paul D. Hanna, <a href="/A300993/b300993.txt">Table of n, a(n) for n = 1..300</a>
%F O.g.f. A(x) satisfies: [x^n] exp( 6*n * A(x) ) = (n + 5) * [x^(n-1)] exp( 6*n * A(x) ) for n>=1.
%F a(n) ~ c * n! * n^11, where c = 0.00000000002970897246102814... - _Vaclav Kotesovec_, Mar 20 2018
%e O.g.f.: A(x) = x + x^2 + 8*x^3 + 84*x^4 + 1080*x^5 + 16056*x^6 + 266256*x^7 + 4816080*x^8 + 93638016*x^9 + 1937252160*x^10 + ...
%e where
%e A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2233*x^4/4! + 141201*x^5/5! + 12458731*x^6/6! + 1435102663*x^7/7! + 206465053425*x^8/8! + 35963535971233*x^9/9! + ... + A300992(n)*x^n/n! + ...
%e A'(x) = 1 + 2*x + 24*x^2 + 336*x^3 + 5400*x^4 + 96336*x^5 + 1863792*x^6 + 38528640*x^7 + 842742144*x^8 + 19372521600*x^9 + ...
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = x*(1-5*x*A')/(1-6*x*A' +x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) /* [x^n] exp( 6*n * A(x) ) = (n + 5) * [x^(n-1)] exp( 6*n * A(x) ) */
%o {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(6*(#A-1))); A[#A] = ((#A+4)*V[#A-1] - V[#A])/(6*(#A-1)) ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A300992, A088716, A300736, A300987, A300989, A300991, A300993.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 19 2018
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