A393864 - OEIS

A393864

O.g.f. A(x) satisfies: [x^n] exp( n^6*x - n^4*A(x) ) = 0 for n >= 1.

%I #10 May 18 2026 04:47:47

%S 1,72,513216,32888512512,9533482907040000,8847761079996449869824,

%T 21208372916660000240980156416,113213233958373606470630646183100416,

%U 1207239247177959014882917182089160500772864,23666053579636080857947549887288559867984281600000,798864336553999630278687419827388750256212948228373479424

%N O.g.f. A(x) satisfies: [x^n] exp( n^6*x - n^4*A(x) ) = 0 for n >= 1.

%H Paul D. Hanna, <a href="/A393864/b393864.txt">Table of n, a(n) for n = 1..201</a>

%F a(n) ~ sqrt(1-c) * 6^(6*n - 4) * n^(5*n - 9/2) / (sqrt(2*Pi) * (6-c)^(5*n - 4) * c^n * exp(5*n)), where c = -LambertW(-6*exp(-6)). - _Vaclav Kotesovec_, May 18 2026

%e O.g.f.: A(x) = x + 72*x^2 + 513216*x^3 + 32888512512*x^4 + 9533482907040000*x^5 + 8847761079996449869824*x^6 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp( n^6*x - n^4*A(x) ) begins

%e n = 1: [1, 0, -144, -3079296, ...];

%e n = 2: [1, 48, 0, -49489920, ...];

%e n = 3: [1, 648, 408240, 0, ...];

%e n = 4: [1, 3840, 14708736, 55410130944, 0, ...];

%e n = 5: [1, 15000, 224910000, 3369025440000, 49894723012320000, 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 + 145*x^2/2! + 3079729*x^3/3! + 789336680545*x^4/4! + 1144021899931593121*x^5/5! + 6370394843516764679184241*x^6/6! + ...

%e where [x^n] exp(n^6*x) / exp(A(x))^(n^4) = 0 for n >= 1.

%o (PARI) {a(n,p=6,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) ); polcoeff( Ser(A), n)}

%o for(n=1, 15, print1(a(n,6,4), ", "))

%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. A393861 (6,1), A393862 (6,2), A393863 (6,3), A393865 (6,5).

%Y Cf. A393854 (5,4), A393874 (7,4), A393884 (8,4).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 02 2026