A393873 O.g.f. A(x) satisfies: [x^n] exp( n^7*x - n^3*A(x) ) = 0 for n >= 1.

1, 900, 60264000, 45080435520000, 188707522069890000000, 3025031182134203198563200000, 145625620250012205493377069120000000, 17775986383636554167912846884425891840000000, 4858544155753756234423057635884960803079755200000000, 2703480132482393823816192151039751243854085358235200000000000

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 - 3) * n^(6*n - 7/2) / (sqrt(2*Pi) * (7-c)^(6*n - 3) * c^n * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 18 2026

Example

O.g.f.: A(x) = x + 900*x^2 + 60264000*x^3 + 45080435520000*x^4 + 188707522069890000000*x^5 + 3025031182134203198563200000*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^7*x - n^3*A(x) ) begins
  n = 1: [1,     0,      -1800,      -361584000, ...];
  n = 2: [1,   120,          0,     -2896128000, ...];
  n = 3: [1,  2160,    4617000,               0, ...];
  n = 4: [1, 16320,  266227200,   4317926400000, 0, ...];
  n = 5: [1, 78000, 6083775000, 474454152000000, 36857499669315000000, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 1801*x^2/2! + 361589401*x^3/3! + 1081931908546801*x^4/4! + 22644908064551238858001*x^5/5! + 2178022587036578013496540707001*x^6/6! + ...
where [x^n] exp(n^7*x) / exp(A(x))^(n^3) = 0 for n >= 1.

Prog

(PARI) {a(n, p=7, 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, 7, 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. A393871 (7,1), A393872 (7,2), A393874 (7,4), A393875 (7,5), A393876 (7,6).

Cf. A319943 (4,3), A393853 (5,3), A393863 (6,3), A393883 (8,3).

Sequence in context: A093798 A093218 A093215 * A252674 A158407 A250782

Adjacent sequences: A393870 A393871 A393872 * A393874 A393875 A393876

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved