A323311 - OEIS

%I #10 Aug 11 2021 17:29:36

%S 1,1,11,283,14855,1310011,172520351,31513669363,7595793146855,

%T 2330879613371851,886383762411615791,408963256168949033443,

%U 225040270250903527024055,145601653678200482159541691,109437844707983885536850408831,94572173789825201408460630621523,93118733370917669491764504635160455,103644400582305503214140030821130959531,129490690058782610512772741408027302955471,180464581077334737195826400036356606725361603

%N E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).

%H Paul D. Hanna, <a href="/A323311/b323311.txt">Table of n, a(n) for n = 0..150</a>

%F E.g.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).

%F (2) 1 = Sum_{n>=0} (exp(n*x) + 1)^n/(A(x) + 1 + exp(n*x))^(n+1).

%F a(n) ~ c * A317904^n * n^(2*n + 1/2) / exp(2*n), where c = 1.5545244013... - _Vaclav Kotesovec_, Aug 11 2021

%e E.g.f.: A(x) = 1 + x + 11*x^2/2! + 283*x^3/3! + 14855*x^4/4! + 1310011*x^5/5! + 172520351*x^6/6! + 31513669363*x^7/7! + 7595793146855*x^8/8! + 2330879613371851*x^9/9! +  + 886383762411615791*x^10/10! + ...

%e such that

%e 1 = 1/A(x)  +  (exp(x) - 1)/(A(x) + 1 - exp(x))^2  +  (exp(2*x) - 1)^2/(A(x) + 1 - exp(2*x))^3  +  (exp(3*x) - 1)^3/(A(x) + 1 - exp(3*x))^4  +  (exp(4*x) - 1)^4/(A(x) + 1 - exp(4*x))^5  +  (exp(5*x) - 1)^5/(A(x) + 1 - exp(5*x))^6 + ...

%e also,

%e 1 = 1/(A(x) + 2)  +  (exp(x) + 1)/(A(x) + 1 + exp(x))^2  +  (exp(2*x) + 1)^2/(A(x) + 1 + exp(2*x))^3  +  (exp(3*x) + 1)^3/(A(x) + 1 + exp(3*x))^4  +  (exp(4*x) + 1)^4/(A(x) + 1 + exp(4*x))^5  +  (exp(5*x) + 1)^5/(A(x) + 1 + exp(5*x))^6 + ...

%e RELATED SERIES.

%e log(A(x)) = x + 10*x^2/2! + 252*x^3/3! + 13486*x^4/4! + 1213260*x^5/5! + 162204670*x^6/6! + 29956649772*x^7/7! + 7279075598686*x^8/8! + 2247264600871500*x^9/9! + ...

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

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A323313.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 02 2019