A393852 O.g.f. A(x) satisfies: [x^n] exp( n^5*x - n^2*A(x) ) = 0 for n >= 1.

1, 98, 214344, 2422736288, 86404408885000, 7522353279784847520, 1361020067428146706895552, 457729018890814234481125167104, 263558187228906049449501760761461088, 243920083465149722342720826648941363212800, 345154327291523887411720506126476368087604332352, 717088358317356048916029332387642017554036276257129472

Offset

1,2

Links

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

Formula

a(n) ~ sqrt(1-c) * 5^(5*n - 2) * n^(4*n - 5/2) / (sqrt(2*Pi) * (5-c)^(4*n-2) * c^n * exp(4*n)), where c = -LambertW(-5*exp(-5)) = 0.0348857682557236963012408686771... - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 98*x^2 + 214344*x^3 + 2422736288*x^4 + 86404408885000*x^5 + 7522353279784847520*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^5*x - n^2*A(x) ) begins
  n = 1: [1,    0,    -196,    -1286064, ...];
  n = 2: [1,   28,       0,    -5188160, ...];
  n = 3: [1,  234,   52992,           0, ...];
  n = 4: [1, 1008, 1012928,   994132224, 0, ...];
  n = 5: [1, 3100, 9605100, 29713278400, 90217316417200, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 197*x^2/2! + 1286653*x^3/3! + 58150931593*x^4/4! + 10368822328678841*x^5/5! + 5416156760994815588941*x^6/6! +  ...
where [x^n] exp(n^5*x) / exp(A(x))^(n^2) = 0 for n >= 1.

Prog

(PARI) {a(n, p=5, q=2) = 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, 5, 2), ", "))

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. A319944 (5,1), A393853 (5,3), A393854 (5,4).

Cf. A317346 (3,2), A319942 (4,2), A393862 (6,2), A393872 (7,2), A393882 (8,2).

Sequence in context: A146199 A159381 A295191 * A295184 A295186 A097601

Adjacent sequences: A393849 A393850 A393851 * A393853 A393854 A393856

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved