A393864 O.g.f. A(x) satisfies: [x^n] exp( n^6*x - n^4*A(x) ) = 0 for n >= 1.

1, 72, 513216, 32888512512, 9533482907040000, 8847761079996449869824, 21208372916660000240980156416, 113213233958373606470630646183100416, 1207239247177959014882917182089160500772864, 23666053579636080857947549887288559867984281600000, 798864336553999630278687419827388750256212948228373479424

Offset

1,2

Links

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

Formula

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

Example

O.g.f.: A(x) = x + 72*x^2 + 513216*x^3 + 32888512512*x^4 + 9533482907040000*x^5 + 8847761079996449869824*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^6*x - n^4*A(x) ) begins
  n = 1: [1,     0,      -144,      -3079296, ...];
  n = 2: [1,    48,         0,     -49489920, ...];
  n = 3: [1,   648,    408240,             0, ...];
  n = 4: [1,  3840,  14708736,   55410130944, 0, ...];
  n = 5: [1, 15000, 224910000, 3369025440000, 49894723012320000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
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! + ...
where [x^n] exp(n^6*x) / exp(A(x))^(n^4) = 0 for n >= 1.

Prog

(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)}
for(n=1, 15, print1(a(n, 6, 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. A393861 (6,1), A393862 (6,2), A393863 (6,3), A393865 (6,5).

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

Sequence in context: A135320 A135427 A028671 * A261815 A159440 A308586

Adjacent sequences: A393861 A393862 A393863 * A393865 A393866 A393867

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved