OFFSET
0,3
COMMENTS
Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,
(2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.
It is remarkable that this sequence should consist entirely of integers.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
a(n) ~ c * n!^3, where c = 13.46489329292094724950380929883219... - Vaclav Kotesovec, Oct 06 2020
EXAMPLE
O.g.f.: A(x) = 1 + x + 20*x^2 + 918*x^3 + 80032*x^4 + 12042925*x^5 + 2930093028*x^6 + 1091180685420*x^7 + 593430683068672*x^8 + 453081063936151719*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (n + 1 - A(x)) begins:
n=0: [0, -1, -40, -5508, -1920768, -1445151000, -2109666980160, ...];
n=1: [1, 0, -39, -5510, -1921491, -1445365884, -2109780457715, ...];
n=2: [2, 7, 0, -4780, -1823168, -1405023192, -2074130121472, ...];
n=3: [3, 26, 239, 0, -1391649, -1249241538, -1942417653741, ...];
n=4: [4, 63, 1080, 21916, 0, -860673816, -1637736990272, ...];
n=5: [5, 124, 3285, 101342, 4459057, 0, -1050171876535, ...];
n=6: [6, 215, 8096, 338580, 18744384, 1958675496, 0, ...];
n=7: [7, 342, 17355, 946660, 61910307, 6852230778, 1865443733743, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 127*x^3/3! + 22537*x^4/4! + 9717681*x^5/5! + 8729681611*x^6/6! + 14829069291583*x^7/7! + 44115361026430737*x^8/8! + ...
PROG
(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^2*x*(Ser(A)) ) * ((m-1)+1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 26 2018
STATUS
approved