A305111 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n - A(x)) = 0 for n >= 0.

1, 0, 2, 15, 232, 5335, 175416, 7847665, 460083056, 34295632587, 3171572232860, 356568882856505, 47917706485943520, 7587855756759594511, 1398620339344282321232, 296926334953470628671285, 71938929865159851175220384, 19730017662046372201511846043, 6081674963212598041745828898948, 2093499393528764015996786711595373, 800173558940733992481330233586267600

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

Table of n, a(n) for n=0..20.

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

Cf. A305110, A305112, A305113, A305114.

Sequence in context: A247660 A197236 A097628 * A216860 A161968 A294043

Adjacent sequences: A305108 A305109 A305110 * A305112 A305113 A305114

Keyword

nonn

Author

Paul D. Hanna, May 26 2018

Status

approved