A305112 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 2 - A(x)) = 0 for n > 0.

1, 2, 14, 195, 4352, 140995, 6253116, 364374941, 27036713728, 2491365948867, 279309067129600, 37451344536549537, 5919866667150287400, 1089585227694809088555, 231042701351773513301428, 55921445108135379302601345, 15324373684519468726237630144, 4720359779903996985477019670883, 1623926325952537873282845134083944, 620377226042860947110418959221630037, 261811933632115596304376828903887527560

Offset: 0

Author

Paul D. Hanna, May 26 2018

Keywords

• nonn

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.

Examples

			O.g.f.: A(x) = 1 + 2*x + 14*x^2 + 195*x^3 + 4352*x^4 + 140995*x^5 + 6253116*x^6 + 364374941*x^7 + 27036713728*x^8 + 2491365948867*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 2 - A(x)) begins:
n=0: [1, -2, -28, -1170, -104448, -16919400, -4502243520, ...];
n=1: [2, 0, -22, -1090, -100566, -16477948, -4412412190, ...];
n=2: [3, 4, 0, -738, -84304, -14795544, -4094683392, ...];
n=3: [4, 10, 44, 0, -52212, -11661618, -3524135580, ...];
n=4: [5, 18, 116, 1262, 0, -6827560, -2672928448, ...];
n=5: [6, 28, 222, 3210, 77582, 0, -1509882570, ...];
n=6: [7, 40, 368, 6030, 186864, 9166632, 0, ...];
n=7: [8, 54, 560, 9932, 335376, 21078266, 1896079208, 0, ...]; ....
in which the main diagonal is all zeros after the initial term, illustrating the  property that [x^n] exp( n * x*A(x) ) * (n + 2 - A(x)) = 0 for n > 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 97*x^3/3! + 5089*x^4/4! + 550121*x^5/5! + 105094981*x^6/6! + 32297760265*x^7/7! + 14961684191777*x^8/8! + ...
		

Crossrefs

Cf. A305110, A305111, A305113, A305114.

Programs

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 - Ser(A)) )[m] );A[n+1]}
for(n=0,20,print1(a(n),", "))