A393854 O.g.f. A(x) satisfies: [x^n] exp( n^5*x - n^4*A(x) ) = 0 for n >= 1.

1, 8, 7452, 48548864, 1136677600000, 70034986070462592, 9435257474951160487152, 2453047052427327473993580544, 1124092061740721156975888283516384, 847407328075222230380235066124690880000, 995470268256052077384803840592163060415989504, 1744311450714495627539492241068129870998549358936064

Offset

1,2

Comments

In general, if p > q >= 1 and o.g.f. A(x) satisfies: [x^n] exp(n^p*x - n^q*A(x)) = 0, then a(n) ~ sqrt(1-c) * p^(p*n - q - s/p) * n^((p-1)*n - q - 1/2) / (sqrt(2*Pi) * (p-c)^((p-1)*n - q) * c^(n - s/p) * exp((p-1)*n)), where c = -LambertW(-p*exp(-p)) and s = 0 if p > q+1 and s = 1 if p = q+1. - Vaclav Kotesovec, May 18 2026

Links

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

Formula

a(n) ~ sqrt(1-c) * 5^(5*n - 21/5) * n^(4*n - 9/2) / (sqrt(2*Pi) * (5-c)^(4*n - 4) * c^(n - 1/5) * exp(4*n)), where c = -LambertW(-5*exp(-5)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 8*x^2 + 7452*x^3 + 48548864*x^4 + 1136677600000*x^5 + 70034986070462592*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^5*x - n^4*A(x) ) begins
  n = 1: [1,    0,     -16,      -44712, ...];
  n = 2: [1,   16,       0,     -723584, ...];
  n = 3: [1,  162,   24948,           0, ...];
  n = 4: [1,  768,  585728,   432101376, 0, ...];
  n = 5: [1, 2500, 6240000, 15522055000, 37680117040000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 17*x^2/2! + 44761*x^3/3! + 1165352449*x^4/4! + 136407145468721*x^5/5! + 50426008695759634321*x^6/6! + ...
where [x^n] exp(n^5*x) / exp(A(x))^(n^4) = 0 for n >= 1.

Prog

(PARI) {a(n, p=5, q=4) = 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, 5, 4), ", "))

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. A319944 (5,1), A393852 (5,2), A393853 (5,3).

Cf. A393864 (6,4), A393874 (7,4), A393884 (8,4).

Cf. A317344 (2,1), A317346 (3,2), A319943 (4,3), A393865 (6,5), A393876 (7,6), A393887 (8,7).

Sequence in context: A356083 A278854 A115442 * A216936 A090494 A079656

Adjacent sequences: A393851 A393852 A393853 * A393856 A393857 A393858

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved