A393883 O.g.f. A(x) satisfies: [x^n] exp( n^8*x - n^3*A(x) ) = 0 for n >= 1.

1, 3844, 1696840596, 11843421298181888, 600567913457231735114000, 143229672214318101538770307845696, 121508290214370553374787406992716549713808, 301933033756295921904657612229325808594540831678464, 1905132525799732817123296403829577016004628248111568229689440

Offset

1,2

Links

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

Formula

a(n) ~ sqrt(1-c) * 8^(8*n - 3) * n^(7*n - 7/2) / (sqrt(2*Pi) * (8-c)^(7*n - 3) * c^n * exp(7*n)), where c = -LambertW(-8*exp(-8)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 3844*x^2 + 1696840596*x^3 + 11843421298181888*x^4 + 600567913457231735114000*x^5 + 143229672214318101538770307845696*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^8*x - n^3*A(x) ) begins
  n = 1: [1,      0,        -7688,      -10181043576, ...];
  n = 2: [1,    248,            0,      -81478854592, ...];
  n = 3: [1,   6534,     42485580,                 0, ...];
  n = 4: [1,  65472,   4286090752,   279902918287872, 0, ...];
  n = 5: [1, 390500, 152489289000, 59545044183053000, 23214878976398803336000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 7689*x^2/2! + 10181066641*x^3/3! + 284242152057901777*x^4/4! + 72068151036861185322722241*x^5/5! + 103125364426751750402197988522653081*x^6/6! + ...
where [x^n] exp(n^8*x) / exp(A(x))^(n^3) = 0 for n >= 1.

Prog

(PARI) {a(n, p=8, q=3) = 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)}
for(n=1, 15, print1(a(n, 8, 3), ", "))

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. A393881 (8,1), A393882 (8,2), A393884 (8,4), A393885 (8,5), A393886 (8,6), A393887 (8,7).

Cf. A319943 (4,3), A393853 (5,3), A393863 (6,3), A393873 (7,3).

Sequence in context: A253291 A253161 A185773 * A031560 A180505 A252878

Adjacent sequences: A393880 A393881 A393882 * A393884 A393885 A393886

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved