A393874 O.g.f. A(x) satisfies: [x^n] exp( n^7*x - n^4*A(x) ) = 0 for n >= 1.

1, 392, 18393804, 10664500259840, 36365399284006880000, 490848817029959167266614400, 20384134822754163932021864236810768, 2186459208756577894213241878174462235443200, 532784557325373969612796632659206881585175523881824, 267393569492843693198477205990632813610787215943794096960000

Offset

1,2

Links

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

Formula

a(n) ~ sqrt(1-c) * 7^(7*n - 4) * n^(6*n - 9/2) / (sqrt(2*Pi) * (7-c)^(6*n - 4) * c^n * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 392*x^2 + 18393804*x^3 + 10664500259840*x^4 + 36365399284006880000*x^5 + 490848817029959167266614400*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^7*x - n^4*A(x) ) begins
  n = 1: [1,     0,       -784,      -110362824, ...];
  n = 2: [1,   112,          0,     -1768615040, ...];
  n = 3: [1,  2106,    4371732,               0, ...];
  n = 4: [1, 16128,  259911680,   4157128783872, 0, ...];
  n = 5: [1, 77500, 6005760000, 465301473235000, 35876031106752400000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 785*x^2/2! + 110365177*x^3/3! + 255948449536129*x^4/4! + 4363849194687214176881*x^5/5! + 353411174447793677420489825041*x^6/6! + ...
where [x^n] exp(n^7*x) / exp(A(x))^(n^4) = 0 for n >= 1.

Prog

(PARI) {a(n, p=7, 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) ); polcoef( Ser(A), n)}
for(n=1, 15, print1(a(n, 7, 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. A393871 (7,1), A393872 (7,2), A393873 (7,3), A393875 (7,5), A393876 (7,6).

Cf. A393854 (5,4), A393864 (6,4), A393884 (8,4).

Sequence in context: A107547 A068276 A068288 * A085019 A105233 A048129

Adjacent sequences: A393871 A393872 A393873 * A393875 A393876 A393877

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved