A393871 O.g.f. A(x) satisfies [x^n] exp( n^7*x - n*A(x) ) = 0 for n >= 1.

1, 3969, 570074232, 739996964038962, 4789222982357994715500, 110008722838279295800818830376, 7189668506900450668818040146415064448, 1144558236708801000444044165482264454510610384, 395552782755203750373573403948089150980508886563735264, 271556424746178362802719627609278241040403769862937059742830480

Offset

1,2

Links

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

Formula

B(x) = x + (1/(m+1)) * A((m+1)^7 * x^(m+1)) satisfies [x^n] exp(n*x*(1+n^6*x^m) - n*B(x) ) = 0 for n >= 1. - Seiichi Manyama, Apr 09 2026

a(n) ~ sqrt(1-c) * 7^(7*n - 1) * n^(6*n - 3/2) / (sqrt(2*Pi) * (7-c)^(6*n - 1) * c^n * exp(6*n)), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, Apr 09 2026

Example

O.g.f.: A(x) = x + 3969*x^2 + 570074232*x^3 + 739996964038962*x^4 + 4789222982357994715500*x^5 + 110008722838279295800818830376*x^6 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^7*x - n*A(x) ) begins
  n = 1: [1,     0,      -7938,     -3420445392, ...];
  n = 2: [1,   126,          0,     -6844891536, ...];
  n = 3: [1,  2184,    4746042,               0, ...];
  n = 4: [1, 16380,  268272648,   4379583997152, 0, ...];
  n = 5: [1, 78120, 6102694710, 476719207352640, 37147770116934096060, 0, ...]; ...
in which a diagonal, the coefficient of x^n in row n, is all zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 7939*x^2/2! + 3420469207*x^3/3! + 17759941007799817*x^4/4! + 574706846954145155463361*x^5/5! + 79206283894033577300908004448571*x^6/6! + ...
where [x^n] exp(n^7*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, p=7, q=1) = 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, 1), ", "))

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. A393872 (7,2), A393873 (7,3), A393874 (7,4), A393875 (7,5), A393876 (7,6).

Cf. A317344 (2,1), A317347 (3,1), A319941 (4,1), A319944 (5,1), A393861 (6,1), A393881 (8,1).

Sequence in context: A343024 A343002 A230067 * A031561 A031741 A204501

Adjacent sequences: A393868 A393869 A393870 * A393872 A393873 A393874

Keyword

nonn

Author

Paul D. Hanna, Mar 02 2026

Status

approved