A393863 O.g.f. A(x) satisfies: [x^n] exp( n^6*x - n^3*A(x) ) = 0 for n >= 1.

1, 196, 1997892, 162416882432, 56765045709658000, 61618456433240675155008, 169099603099382087617231204528, 1016837678287497214761101393803075584, 12059474305653939620072795894420421876287328, 260238444138573786264768893486595871180295992224000, 9588454992266710212994015747538592832746521658797171857536

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..201

Formula

a(n) ~ sqrt(1-c) * 6^(6*n - 3) * n^(5*n - 7/2) / (sqrt(2*Pi) * (6-c)^(5*n - 3) * c^n * exp(5*n)), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 196*x^2 + 1997892*x^3 + 162416882432*x^4 + 56765045709658000*x^5 + 61618456433240675155008*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^6*x - n^3*A(x) ) begins
  n = 1: [1,     0,      -392,     -11987352, ...];
  n = 2: [1,    56,         0,     -96250048, ...];
  n = 3: [1,   702,    482220,             0, ...];
  n = 4: [1,  4032,  16231936,   64477665792, 0, ...];
  n = 5: [1, 15500, 240201000, 3720098081000, 57069283577704000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 393*x^2/2! + 11988529*x^3/3! + 3898053591121*x^4/4! + 6811825022297454081*x^5/5! + 44365329527182186764295321*x^6/6! + ...
where [x^n] exp(n^6*x) / exp(A(x))^(n^3) = 0 for n >= 1.

Prog

(PARI) {a(n, p=6, q=3) = my(A=[0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^p*x - m^q*Ser(A)) )[m+1]/(m^q) ); polcoef( Ser(A), n)}
for(n=1, 15, print1(a(n, 6, 3), ", "))

Crossrefs

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. A393861 (6,1), A393862 (6,2), A393864 (6,4), A393865 (6,5).

Cf. A319943 (4,3), A393853 (5,3), A393873 (7,3), A393883 (8,3).

Sequence in context: A351683 A353064 A013755 * A071410 A013871 A306856

Adjacent sequences: A393860 A393861 A393862 * A393864 A393865 A393866

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved