A393875 O.g.f. A(x) satisfies: [x^n] exp( n^7*x - n^5*A(x) ) = 0 for n >= 1.

1, 144, 4758912, 2174850957312, 6141506309712000000, 70785775509920057340297216, 2565634975195302147272897051394048, 244187652276693464916674766704731458895872, 53484631047625356847546933903612842552393213149184, 24380091982714424468767337810324545495755967688343552000000

Offset

1,2

Links

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

Formula

a(n) ~ sqrt(1-c) * 7^(7*n - 5) * n^(6*n - 11/2) / (sqrt(2*Pi) * (7-c)^(6*n - 5) * c^n * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 144*x^2 + 4758912*x^3 + 2174850957312*x^4 + 6141506309712000000*x^5 + 70785775509920057340297216*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^7*x - n^5*A(x) ) begins
  n = 1: [1,     0,       -288,       -28553472, ...];
  n = 2: [1,    96,          0,      -915480576, ...];
  n = 3: [1,  1944,    3709152,               0, ...];
  n = 4: [1, 15360,  235634688,   3581050355712, 0, ...];
  n = 5: [1, 75000, 5624100000, 421583270400000, 31420369728201600000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 289*x^2/2! + 28554337*x^3/3! + 52196537439937*x^4/4! + 736981018230075658561*x^5/5! + 50965762789261409273222404321*x^6/6! + ...
where [x^n] exp(n^7*x) / exp(A(x))^(n^5) = 0 for n >= 1.

Prog

(PARI) {a(n, p=7, q=5) = 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, 7, 5), ", "))

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. A393871 (7,1), A393872 (7,2), A393873 (7,3), A393874 (7,4), A393876 (7,6).

Cf. A393865 (6,5), A393885 (8,5).

Sequence in context: A159436 A318197 A362214 * A193346 A003837 A013863

Adjacent sequences: A393872 A393873 A393874 * A393876 A393877 A393878

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved