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) * 7^(7*n - 43/7) * n^(6*n - 13/2) / (sqrt(2*Pi) * (7-c)^(6*n - 6) * c^(n - 1/7) * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 18 2026
EXAMPLE
O.g.f.: A(x) = x + 32*x^2 + 661932*x^3 + 221380591616*x^4 + 492195926802500000*x^5 + 4671419955175042830752256*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^7*x - n^6*A(x) ) begins
n = 1: [1, 0, -64, -3971592, ...];
n = 2: [1, 64, 0, -254706176, ...];
n = 3: [1, 1458, 2079108, 0, ...];
n = 4: [1, 12288, 150732800, 1829494554624, 0, ...];
n = 5: [1, 62500, 3905250000, 243891068875000, 15136822809394000000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 65*x^2/2! + 3971785*x^3/3! + 5313150097825*x^4/4! + 59063537784552590801*x^5/5! + 3363422722112436192550045921*x^6/6! + ...
where [x^n] exp(n^7*x) / exp(A(x))^(n^6) = 0 for n >= 1.
PROG
(PARI) {a(n, p=7, q=6) = 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, 6), ", "))
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. A393886 (8,6).
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 02 2026
STATUS
approved