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A305110 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 1 - A(x)) = 0 for n >= 0. 16
1, 1, 6, 69, 1292, 35795, 1381662, 71137787, 4723263400, 393582311685, 40255365257750, 4961198428428121, 725404352867194164, 124184747391389562735, 24610467350470767598014, 5590555396845759790005015, 1443198598348337621150240912, 420171257194869987686583296425, 137030366263275866352884688085734, 49758716350139864002666237593244717, 20008715354387819486054019707449804060 (list; graph; refs; listen; history; text; internal format)
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!^2, where c = 3.73613614872360588359758608879980516876... - Vaclav Kotesovec, Oct 06 2020

EXAMPLE

O.g.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 1292*x^4 + 35795*x^5 + 1381662*x^6 + 71137787*x^7 + 4723263400*x^8 + 393582311685*x^9 + 40255365257750*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 1 - A(x)) begins:

n=0: [0, -1, -12, -414, -31008, -4295400, -994796640, ...];

n=1: [1, 0, -11, -416, -31227, -4312664, -996840335, ...];

n=2: [2, 3, 0, -302, -27216, -3971736, -942218944, ...];

n=3: [3, 8, 27, 0, -17481, -3201216, -823857561, ...];

n=4: [4, 15, 76, 586, 0, -1913384, -633716000, ...];

n=5: [5, 24, 153, 1576, 27897, 0, -362610715, ...];

n=6: [6, 35, 264, 3114, 69648, 2672616, 0, ...];

n=7: [7, 48, 415, 5368, 129579, 6269224, 466272571, 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! + 43*x^3/3! + 1825*x^4/4! + 164481*x^5/5! + 26795491*x^6/6! + 7156366603*x^7/7! + 2926639231713*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 - Ser(A)) )[m] ); A[n+1]}

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

CROSSREFS

Cf. A305111, A305112, A305113, A305114.

Sequence in context: A338176 A235327 A098639 * A235328 A296783 A218683

Adjacent sequences:  A305107 A305108 A305109 * A305111 A305112 A305113

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 25 2018

STATUS

approved

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Last modified September 22 11:28 EDT 2021. Contains 347606 sequences. (Running on oeis4.)