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%I #22 Mar 20 2018 10:32:37
%S 1,1,5,36,327,3489,42048,559008,8073243,125328411,2075525505,
%T 36460943208,676484058564,13210384019292,270753854165604,
%U 5810388957096552,130292809125319539,3047472204302259711,74227110587569392471,1879966895740420683492,49443968787368161215087,1348661750106914651234385,38107004920979745293594856,1114125483618428275543280400,33669232396216806674333898900
%N O.g.f. A(x) satisfies: A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).
%C O.g.f. equals the logarithm of the e.g.f. of A300986.
%C The e.g.f. G(x) of A300986 satisfies: [x^n] G(x)^(3*n) = (n+2) * [x^(n-1)] G(x)^(3*n) for n>=1.
%H Paul D. Hanna, <a href="/A300987/b300987.txt">Table of n, a(n) for n = 1..200</a>
%F O.g.f. A(x) satisfies: [x^n] exp( 3*n * A(x) ) = (n + 2) * [x^(n-1)] exp( 3*n * A(x) ) for n>=1.
%F a(n) ~ c * n! * n^5, where c = 0.00014640560804... - _Vaclav Kotesovec_, Mar 20 2018
%e O.g.f.: A(x) = x + x^2 + 5*x^3 + 36*x^4 + 327*x^5 + 3489*x^6 + 42048*x^7 + 559008*x^8 + 8073243*x^9 + 125328411*x^10 + 2075525505*x^11 + ...
%e where
%e A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1009*x^4/4! + 44541*x^5/5! + 2799931*x^6/6! + 233188033*x^7/7! + 24562692897*x^8/8! + 3168510747769*x^9/9! + 488856473079571*x^10/10! + ... + A300986(n)*x^n/n! + ...
%e A'(x) = 1 + 2*x + 15*x^2 + 144*x^3 + 1635*x^4 + 20934*x^5 + 294336*x^6 + 4472064*x^7 + 72659187*x^8 + 1253284110*x^9 + 22830780555*x^10 + ...
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = x*(1-2*x*A')/(1-3*x*A' +x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) /* [x^n] exp( 3*n * A(x) ) = (n + 2) * [x^(n-1)] exp( 3*n * A(x) ) */
%o {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(3*(#A-1))); A[#A] = ((#A+1)*V[#A-1] - V[#A])/(3*(#A-1)) ); polcoeff( log(Ser(A)), n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A300986, A088716, A300736, A300989, A300991, A300993.
%Y Cf. A296171, A300593, A300595.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 17 2018
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