A323311 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).

1, 1, 11, 283, 14855, 1310011, 172520351, 31513669363, 7595793146855, 2330879613371851, 886383762411615791, 408963256168949033443, 225040270250903527024055, 145601653678200482159541691, 109437844707983885536850408831, 94572173789825201408460630621523, 93118733370917669491764504635160455, 103644400582305503214140030821130959531, 129490690058782610512772741408027302955471, 180464581077334737195826400036356606725361603

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..150

Formula

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

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

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

a(n) ~ c * A317904^n * n^(2*n + 1/2) / exp(2*n), where c = 1.5545244013... - Vaclav Kotesovec, Aug 11 2021

Example

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! + ...
such that
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 + ...
also,
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 + ...
RELATED SERIES.
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! + ...

Prog

(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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323313.

Sequence in context: A103547 A171195 A274780 * A355428 A280359 A196790

Adjacent sequences: A323308 A323309 A323310 * A323312 A323313 A323314

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2019

Status

approved