OFFSET
0,3
COMMENTS
Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
(2) [x^n] exp( x*F(x) ) * (n + 1 - 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..300
FORMULA
a(n) ~ c * n!^3, where c = 2.49393609789981559563078907122202821077556480458411... - Vaclav Kotesovec, Oct 06 2020
EXAMPLE
O.g.f.: A(x) = 1 + x + 14*x^2 + 450*x^3 + 31144*x^4 + 4041775*x^5 + 890769366*x^6 + 309205147860*x^7 + 159530833094816*x^8 + 116905524905145753*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n^2 + 1 - A(x)) begins:
n=0: [0, -1, -28, -2700, -747456, -485013000, -641353943520, ...];
n=1: [1, 0, -27, -2702, -747963, -485118684, -641396951615, ...];
n=2: [4, 7, 0, -2092, -678784, -462055752, -623679177536, ...];
n=3: [9, 26, 101, 0, -460275, -391250658, -569892209247, ...];
n=4: [16, 63, 348, 4828, 0, -246538056, -461135488928, ...];
n=5: [25, 124, 837, 14150, 810509, 0, -277891671695, ...];
n=6: [36, 215, 1688, 30348, 2099712, 378224376, 0, ...];
n=7: [49, 342, 3045, 56548, 4020741, 920163738, 393372598609, 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! + 91*x^3/3! + 11161*x^4/4! + 3793881*x^5/5! + 2933070331*x^6/6! + 4510118566003*x^7/7! + 12503335235913201*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)*x*(Ser(A)) ) * ((m-1)^2+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