A393846 O.g.f. A(x) satisfies: [x^n] exp( n^6 * x ) / A(x) = 0 for n>0.

1, 1, 1985, 62861994, 11442690973075, 7570836550478960487, 13956629370284243750857868, 60197488880602894559278221637956, 536790646696527838745317223865162477075, 9023894622972758071209681557435066872979358504

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

a(n) ~ sqrt(1-c) * 6^(6*n) * n^(5*n - 1/2) / (sqrt(2*Pi) * c^n * (6-c)^(5*n) * exp(5*n)), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = 1 + x + 1985*x^2 + 62861994*x^3 + 11442690973075*x^4 + 7570836550478960487*x^5 + 13956629370284243750857868*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^6*x)/A(x) begins
  n = 1: [1,     0,     -3969,    -377160056, ...];
  n = 2: [1,    63,         0,    -377660150, ...];
  n = 3: [1,   728,    526015,             0, ...];
  n = 4: [1,  4095,  16765056,   68243238154, 0, ...];
  n = 5: [1, 15624, 244105407, 3813401695600, 59285380255397541, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.

Prog

(PARI) {a(n, k=6) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)^k +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 15, print1( a(n, 6), ", "))

Crossrefs

Sequences below with the given parameter p have an o.g.f. F(x) satisfying [x^n] exp( n^p*x ) / F(x) = 0 for n >= 1.

Cf. A304322 (p=2), A304323 (p=3), A304324 (p=4), A304325 (p=5).

Cf. A393847 (p=7), A393848 (p=8), A393849 (p=9).

Sequence in context: A107527 A034228 A028516 * A376831 A206363 A233712

Adjacent sequences: A393843 A393844 A393845 * A393847 A393848 A393849

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved