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..301

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

B(x) = x + (1/(m+1)) * A((m+1)^3 * x^(m+1)) satisfies [x^n] exp(n*x*(1+n^2*x^m) - n*B(x) ) = 0 for n >= 1. - Seiichi Manyama, Apr 09 2026

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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^3*x - n*A(x) ) begins:
  n=1: [1,   0,   -18,   -3312,  -2059236, ...];
  n=2: [1,   6,     0,   -7056,  -4281984, ...];
  n=3: [1,  24,   522,       0,  -6980580, ...];
  n=4: [1,  60,  3528,  189792,         0, ...];
  n=5: [1, 120, 14310, 1679040, 181358460, 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! + ...
where [x^n] exp(n^3*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, k=0) = my(A=[0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*(m+k)*x - m*Ser(A)) )[m+1]/m ); polcoef( Ser(A), n)}
for(n=1, 15, print1(a(n, 0), ", "))

Crossrefs

Cf. A393760, A393761, A393762, A393763, A393764, A393765, A393766, A393767.

Below, sequences with the parameters (p,q) indicated have an o.g.f. F(x) satisfying [x^n] exp( n^p*x - n^q*F(x) ) = 0 for n >= 1.

Cf. A317344 (2,1), this sequence (3,1), A319941 (4,1), A319944 (5,1), A393861 (6,1), A393871 (7,1), A393881 (8,1).

Cf. A317346 (3,2).

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

Extensions

Entry revised by Paul D. Hanna, Mar 01 2026

Status

approved