A393853 O.g.f. A(x) satisfies: [x^n] exp( n^5*x - n^3*A(x) ) = 0 for n >= 1.

1, 36, 54432, 484157952, 14341070088000, 1069388709947831808, 169241367640617997627392, 50577166768732619165669523456, 26201243804894888012465274226741248, 22037908435373887477062906608302823424000, 28578363358880431076573073973885461018380795904, 54795450390474606316958220236698406535473510158958592

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-3) * n^(4*n - 7/2) / (sqrt(2*Pi) * (5-c)^(4*n-3) * 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 + 36*x^2 + 54432*x^3 + 484157952*x^4 + 14341070088000*x^5 + 1069388709947831808*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^5*x - n^3*A(x) ) begins
  n = 1: [1,    0,     -72,     -326592, ...];
  n = 2: [1,   24,       0,    -2640384, ...];
  n = 3: [1,  216,   44712,           0, ...];
  n = 4: [1,  960,  916992,   850563072, 0, ...];
  n = 5: [1, 3000, 8991000, 26878176000, 78571881144000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 73*x^2/2! + 326809*x^3/3! + 11621113201*x^4/4! + 1720986748004881*x^5/5! + 769970210524619827321*x^6/6! + ...
where [x^n] exp(n^5*x) / exp(A(x))^(n^3) = 0 for n >= 1.

Prog

(PARI) {a(n, p=5, 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, 5, 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. A319944 (5,1), A393852 (5,2), A393854 (5,4).

Cf. A319943 (4,3), A393863 (6,3), A393873 (7,3), A393883 (8,3).

Sequence in context: A123397 A185097 A023111 * A295927 A059493 A212327

Adjacent sequences: A393850 A393851 A393852 * A393854 A393856 A393857

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved