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%I #11 Mar 20 2018 10:33:15
%S 1,1,7,66,769,10405,157540,2609120,46569365,886686635,17878075475,
%T 379658470550,8456459003700,196924945517500,4781934837995500,
%U 120830066634026000,3171336593264680125,86326557180541530375,2433904105428495204125,70991247725936793222750,2139844431980825211095625,66589703922949982943093125,2137375830643389417662717500,70701647390083655222041600000
%N O.g.f. A(x) satisfies: A(x) = x * (1 - 4*x*A'(x)) / (1 - 5*x*A'(x)).
%C O.g.f. equals the logarithm of the e.g.f. of A300990.
%C The e.g.f. G(x) of A300990 satisfies: [x^n] G(x)^(5*n) = (n+4) * [x^(n-1)] G(x)^(5*n) for n>=1.
%H Paul D. Hanna, <a href="/A300991/b300991.txt">Table of n, a(n) for n = 1..300</a>
%F O.g.f. A(x) satisfies: [x^n] exp( 5*n * A(x) ) = (n + 4) * [x^(n-1)] exp( 5*n * A(x) ) for n>=1.
%F a(n) ~ c * n! * n^9, where c = 0.00000000758078051425272... - _Vaclav Kotesovec_, Mar 20 2018
%e O.g.f.: A(x) = x + x^2 + 7*x^3 + 66*x^4 + 769*x^5 + 10405*x^6 + 157540*x^7 + 2609120*x^8 + 46569365*x^9 + 886686635*x^10 + ...
%e where
%e A(x) = x * (1 - 4*x*A'(x)) / (1 - 5*x*A'(x)).
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 3*x^2/2! + 49*x^3/3! + 1777*x^4/4! + 101541*x^5/5! + 8140411*x^6/6! + 855134533*x^7/7! + 112545136929*x^8/8! + 17984228218057*x^9/9! + ... + A300990(n)*x^n/n! + ...
%e A'(x) = 1 + 2*x + 21*x^2 + 264*x^3 + 3845*x^4 + 62430*x^5 + 1102780*x^6 + 20872960*x^7 + 419124285*x^8 + 8866866350*x^9 + ...
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = x*(1-4*x*A')/(1-5*x*A' +x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) /* [x^n] exp( 5*n * A(x) ) = (n + 4) * [x^(n-1)] exp( 5*n * A(x) ) */
%o {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(5*(#A-1))); A[#A] = ((#A+3)*V[#A-1] - V[#A])/(5*(#A-1)) ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A300990, A088716, A300736, A300987, A300989, A300993.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 19 2018
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