A393847 O.g.f. A(x) satisfies: [x^n] exp( n^7 * x ) / A(x) = 0 for n>0.

1, 1, 8065, 1723380877, 2972352315820441, 24013530904194819396970, 661513372249008158305805082187, 50423205212193924706520234988903231406, 9172099526697986231143005697761988750382396953, 3565599162605040989126220209472530594315226235614225958

Offset

0,3

Links

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

Formula

a(n) ~ sqrt(1-c) * 7^(7*n) * n^(6*n - 1/2) / (sqrt(2*Pi) * c^n * (7-c)^(6*n) * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = 1 + x + 8065*x^2 + 1723380877*x^3 + 2972352315820441*x^4 + 24013530904194819396970*x^5 + 661513372249008158305805082187*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^7*x)/A(x) begins
  n = 1: [1,     0,     -16129,    -10340236874, ...];
  n = 2: [1,   127,          0,    -10344333640, ...];
  n = 3: [1,  2186,    4762467,               0, ...];
  n = 4: [1, 16383,  268386560,   4386108292792, 0, ...];
  n = 5: [1, 78124, 6103343247, 476804727467762, 37175837330786900381, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.

Prog

(PARI) {a(n, k=7) = 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, 7), ", "))

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. A393846 (p=6), A393848 (p=8), A393849 (p=9).

Sequence in context: A318104 A013378 A013458 * A172775 A172843 A258541

Adjacent sequences: A393844 A393845 A393846 * A393848 A393849 A393850

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved