|
| |
|
|
A323311
|
|
E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - 1)^n/(A(x) + 1 - exp(n*x))^(n+1).
|
|
3
|
|
|
|
1, 1, 11, 283, 14855, 1310011, 172520351, 31513669363, 7595793146855, 2330879613371851, 886383762411615791, 408963256168949033443, 225040270250903527024055, 145601653678200482159541691, 109437844707983885536850408831, 94572173789825201408460630621523, 93118733370917669491764504635160455, 103644400582305503214140030821130959531, 129490690058782610512772741408027302955471, 180464581077334737195826400036356606725361603
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
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).
|
|
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
