A316369 - OEIS

%I #7 Jun 30 2018 19:19:02

%S 1,3,18,2034,1430514,6373869750,199297681460658,46580417624524112586,

%T 84489819173818196151339522,1218062967741942570285262683718734,

%U 141844019201414810730077234469542638090002,134989625848267741768606388581746116482248689575890,1059352182865192441815498125890206907704205904500753040647218

%N O.g.f. A(x) satisfies: Sum_{n>=0} log( (1 + 3^n*x)/A(x) )^n / n!  =  1.

%C It is remarkable that this sequence should consist entirely of integers.

%e O.g.f. A(x) = 1 + 3*x + 18*x^2 + 2034*x^3 + 1430514*x^4 + 6373869750*x^5 + 199297681460658*x^6 + 46580417624524112586*x^7 + ...

%e such that

%e 1  =  1 + log( (1 + 3*x)/A(x) ) + log( (1 + 3^2*x)/A(x) )^2/2! + log( (1 + 3^3*x)/A(x) )^3/3! + log( (1 + 3^4*x)/A(x) )^4/4! + log( (1 + 3^5*x)/A(x) )^5/5! + ... + log( (1 + 3^n*x)/A(x) )^n / n! + ...

%e RELATED SERIES.

%e log(A(x)) = 3*x + 27*x^2/2 + 5967*x^3/3 + 5697567*x^4/4 + 31847802183*x^5/5 + 1195671270431187*x^6/6 + 326058737699333461707*x^7/7 + 675917435446065515610996255*x^8/8 + ... + A316368(n)*x^n/n + ...

%e Sum_{n>=0} 2^n * log( (1 + 3^n*x)/A(x) )^n / n! = 1 + 36*x^2 + 13176*x^3 + 20920896*x^4 + 195419457576*x^5 + 12486795580921716*x^6 + 5897630526144664632768*x^7 + 21506467482283382814751438176*x^8 + ...

%o (PARI) {a(n) = my(A=[3]); for(i=1,n, A=concat(A,0); A[#A] = Vec(sum(n=0,#A+1, (log(1 + 3^n*x +x*O(x^#A) ) - x*Ser(A))^n/n! ))[#A+1]); polcoeff(exp(x*Ser(A)),n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A316368, A306062.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 30 2018