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) * 6^(6*n - 31/6) * n^(5*n - 11/2) / (sqrt(2*Pi) * (6-c)^(5*n - 5) * c^(n - 1/6) * exp(5*n)), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 18 2026
EXAMPLE
O.g.f.: A(x) = x + 16*x^2 + 70956*x^3 + 3330535424*x^4 + 761427167500000*x^5 + 582638218357483190016*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^6*x - n^5*A(x) ) begins
n = 1: [1, 0, -32, -425736, ...];
n = 2: [1, 32, 0, -13689088, ...];
n = 3: [1, 486, 228420, 0, ...];
n = 4: [1, 3072, 9404416, 28253085696, 0, ...];
n = 5: [1, 12500, 156150000, 1948044575000, 24004031093200000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 33*x^2/2! + 425833*x^3/3! + 79934556385*x^4/4! + 91371659904759441*x^5/5! + 419500065486328595531521*x^6/6! + ...
where [x^n] exp(n^6*x) / exp(A(x))^(n^5) = 0 for n >= 1.
PROG
(PARI) {a(n, p=6, 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, 6, 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.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 02 2026
STATUS
approved