A305111 - OEIS

%I #5 Jun 06 2018 20:09:16

%S 1,0,2,15,232,5335,175416,7847665,460083056,34295632587,3171572232860,

%T 356568882856505,47917706485943520,7587855756759594511,

%U 1398620339344282321232,296926334953470628671285,71938929865159851175220384,19730017662046372201511846043,6081674963212598041745828898948,2093499393528764015996786711595373,800173558940733992481330233586267600

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

%C Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:

%C (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,

%C (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.

%C It is remarkable that this sequence should consist entirely of integers.

%e 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 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n - A(x)) begins:

%e n=0: [-1, 0, -4, -90, -5568, -640200, -126299520, ...];

%e n=1: [0, 0, -4, -102, -5952, -669460, -130272180, ...];

%e n=2: [1, 2, 0, -82, -5456, -636888, -126025856, ...];

%e n=3: [2, 6, 14, 0, -3678, -527934, -112372002, ...];

%e n=4: [3, 12, 44, 198, 0, -323848, -87935232, ...];

%e n=5: [4, 20, 96, 590, 6532, 0, -51109520, ...];

%e n=6: [5, 30, 176, 1278, 17328, 476520, 0, ...];

%e n=7: [6, 42, 290, 2388, 34374, 1150022, 67654674, 0, ...];

%e 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.

%o (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]}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A305110, A305112, A305113, A305114.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 26 2018