|
| |
|
|
A298697
|
|
E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)^(2*(n-1)) * x^n/n! / A(x)^((n+1)^2).
|
|
1
|
|
|
|
1, 1, 3, 49, 2461, 234921, 35329471, 7619812873, 2224009220889, 843836219043121, 403933464464241691, 238294863084819823041, 169987570007123851643893, 144326641180625057628760729, 143904971264557527854144891031, 166568515940712946346074697276761, 221584133833606888317064501087446961
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Compare e.g.f. to: 1 = Sum_{n>=0} m * (m*(n+1))^(n-1) * x^n/n! / exp(x)^(m*(n+1)) holds for fixed m.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 49*x^3/3! + 2461*x^4/4! + 234921*x^5/5! + 35329471*x^6/6! + 7619812873*x^7/7! + 2224009220889*x^8/8! + ...
such that
1 = 1/A(x) + x/A(x)^4 + 3^2*x^2/2!/A(x)^9 + 4^4*x^3/3!/A(x)^16 + 5^6*x^4/4!/A(x)^25 + 6^8*x^5/5!/A(x)^36 + 7^10*x^6/6!/A(x)^49 + ...
RELATED SERIES.
As a power series in x with reduced fractional coefficients,
A(x) = 1 + x + 3/2*x^2 + 49/6*x^3 + 2461/24*x^4 + 78307/40*x^5 + 35329471/720*x^6 + 7619812873/5040*x^7 + 741336406963/13440*x^8 + 843836219043121/362880*x^9 + 57704780637748813/518400*x^10 + ...
The logarithm of e.g.f. A(x) begins:
log(A(x)) = x + 2*x^2/2! + 42*x^3/3! + 2268*x^4/4! + 222240*x^5/5! + 33870120*x^6/6! + 7364669760*x^7/7! + 2161424593440*x^8/8! + 823377654610560*x^9/9! + 395340117541407360*x^10/10! ...
|
|
|
MATHEMATICA
|
terms = 17; A[_] = 1; Do[A[x_] = A[x] - 1 + Sum[(n + 1)^(2*(n - 1))*x^n/n!/A[x]^((n+1)^2), {n, 0, terms}] + O[x]^terms, {terms}]; CoefficientList[ A[x], x]*Range[0, terms-1]! (* Jean-François Alcover, Feb 09 2018 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(m=0, #A, ((m+1)^2)^(m-1) * x^m/m! /Ser(A)^((m+1)^2) ))[#A] ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
