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.
EXAMPLE
O.g.f.: A(x) = 1 + 2*x^2 + 15*x^3 + 232*x^4 + 5335*x^5 + 175416*x^6 + 7847665*x^7 + 460083056*x^8 + 34295632587*x^9 + 3171572232860*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n - A(x)) begins:
n=0: [-1, 0, -4, -90, -5568, -640200, -126299520, ...];
n=1: [0, 0, -4, -102, -5952, -669460, -130272180, ...];
n=2: [1, 2, 0, -82, -5456, -636888, -126025856, ...];
n=3: [2, 6, 14, 0, -3678, -527934, -112372002, ...];
n=4: [3, 12, 44, 198, 0, -323848, -87935232, ...];
n=5: [4, 20, 96, 590, 6532, 0, -51109520, ...];
n=6: [5, 30, 176, 1278, 17328, 476520, 0, ...];
n=7: [6, 42, 290, 2388, 34374, 1150022, 67654674, 0, ...];
in which the main diagonal is all zeros after the initial term, illustrating the property that exp( n * x*A(x) ) * (n - A(x)) for n >= 0.
PROG
(PARI) {a(n) = my(A=[1, 0], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * (m-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