A317347 O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n*A(x) ) = 0 for n >= 1.

1, 9, 552, 85842, 24653700, 11219022936, 7393496092416, 6649411839351120, 7822998961379912592, 11662362974001268456560, 21487905123054927319268352, 47958258768575173308988367040, 127523196462392124262710980808384, 398397752352904475778061859746030080, 1445051361690004153927005867189533921280

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

a(n) ~ sqrt(1-c) * 3^(3*n-1) * n^(2*n - 3/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n-1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 06 2018

Example

O.g.f.: A(x) = x + 9*x^2 + 552*x^3 + 85842*x^4 + 24653700*x^5 + 11219022936*x^6 + 7393496092416*x^7 + 6649411839351120*x^8 + ...
such that [x^n] exp( n^3*x - n*A(x) ) = 0  for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients in  begins:
n=1: [1, 0, -18, -3312, -2059236, -2957847840, -8077030651800, ...];
n=2: [1, 6, 0, -7056, -4281984, -6040453824, -16367904244224, ...];
n=3: [1, 24, 522, 0, -6980580, -9667325376, -25560523291464, ...];
n=4: [1, 60, 3528, 189792, 0, -14146669440, -37025599219200, ...];
n=5: [1, 120, 14310, 1679040, 181358460, 0, -51097553724600, ...];
n=6: [1, 210, 43992, 9173088, 1887214464, 358972896960, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 19*x^2/2! + 3367*x^3/3! + 2074537*x^4/4! + 2969379361*x^5/5! + 8096147776171*x^6/6! + 37321188279552199*x^7/7! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A317344, A317346.

Sequence in context: A054608 A269661 A287043 * A357229 A267548 A326614

Adjacent sequences: A317344 A317345 A317346 * A317348 A317349 A317350

Keyword

nonn

Author

Paul D. Hanna, Jul 29 2018

Status

approved