A393764 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+4)*x - n*A(x) ) = 0 for n >= 1.

5, 49, 3792, 726674, 248954740, 131319894744, 97960976774752, 97816828413705040, 125776784844261867600, 202322792207916645666800, 398042334632375210744679680, 940438668044289996907775127744, 2628397838622257811970918101586880, 8579795091273877351227072749607517696, 32355183395235713189688333653980520785920

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..301

Formula

a(n) ~ sqrt(1-w) * 3^(3*n + 1/3) * n^(2*n - 3/2) / (sqrt(2*Pi) * exp(2*n) * (3-w)^(2*n-1) * w^(n + 4/3)), where w = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Mar 02 2026

Example

O.g.f.: A(x) = 5*x + 49*x^2 + 3792*x^3 + 726674*x^4 + 248954740*x^5 + 131319894744*x^6 + 97960976774752*x^7 + 97816828413705040*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+4)*x - n*A(x) ) begin
  n = 1: [1,   0,   -98,  -22752,  -17411364, ...];
  n = 2: [1,  14,     0,  -50992,  -37505408, ...];
  n = 3: [1,  48,  2010,       0,  -63922212, ...];
  n = 4: [1, 108, 11272, 1041696,          0, ...];
  n = 5: [1, 200, 39510, 7592240, 1304911420, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, equals all zeros.
RELATED SERIES.
exp(A(x)) = 1 + 5*x + 123*x^2/2! + 24347*x^3/3! + 17939353*x^4/4! + 30339404085*x^5/5! + 95484666584995*x^6/6! + ...
where [x^n] exp(n^2*(n+4)*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, k=4) = my(A=[0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*(m+k)*x - m*Ser(A)) )[m+1]/m ); polcoef( Ser(A), n)}
for(n=1, 15, print1(a(n, 4), ", "))

Crossrefs

Cf. A393760, A317347, A393761, A393762, A393763, A393765, A393766, A393767.

Sequence in context: A224680 A075986 A251657 * A084765 A203411 A218322

Adjacent sequences: A393761 A393762 A393763 * A393765 A393766 A393767

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved