A393854 - OEIS

A393854

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

%I #14 May 18 2026 06:28:01

%S 1,8,7452,48548864,1136677600000,70034986070462592,

%T 9435257474951160487152,2453047052427327473993580544,

%U 1124092061740721156975888283516384,847407328075222230380235066124690880000,995470268256052077384803840592163060415989504,1744311450714495627539492241068129870998549358936064

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

%C 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

%H Paul D. Hanna, <a href="/A393854/b393854.txt">Table of n, a(n) for n = 1..201</a>

%F 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

%e O.g.f.: A(x) = x + 8*x^2 + 7452*x^3 + 48548864*x^4 + 1136677600000*x^5 + 70034986070462592*x^6 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp( n^5*x - n^4*A(x) ) begins

%e n = 1: [1, 0, -16, -44712, ...];

%e n = 2: [1, 16, 0, -723584, ...];

%e n = 3: [1, 162, 24948, 0, ...];

%e n = 4: [1, 768, 585728, 432101376, 0, ...];

%e n = 5: [1, 2500, 6240000, 15522055000, 37680117040000, 0, ...]; ...

%e in which a diagonal, the coefficient of x^n in row n, is all zeros.

%e RELATED SERIES.

%e 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! + ...

%e where [x^n] exp(n^5*x) / exp(A(x))^(n^4) = 0 for n >= 1.

%o (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)}

%o for(n=1, 15, print1(a(n,5,4), ", "))

%Y 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.

%Y Cf. A319944 (5,1), A393852 (5,2), A393853 (5,3).

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

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

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 02 2026