OFFSET
0,3
COMMENTS
Old name was: A simple grammar.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..310
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 752.
FORMULA
E.g.f.: RootOf(-_Z+_Z^6*x+1).
D-finite recurrence: {a(1)=1, a(2)=12, (-720-9864*n-48600*n^2-110160*n^3-116640*n^4-46656*n^5)*a(n)+(3125*n^4+9375*n^3+10000*n^2+4500*n+720)*a(n+1), a(6)=45239040, a(3)=306, a(4)=12144, a(5)=657720}.
a(n) = (1/25) * 3^(1/2) * (5+5^(1/2))^(1/2) * (5-5^(1/2))^(1/2) * Pi^(1/2) * Gamma(2*n+37/3) * Gamma(2*n+38/3) / Gamma(n+34/5)/Gamma(n+33/5) / Gamma(n+32/5) / Gamma(n+36/5) *GAMMA(n+13/2)*3125^(-6-n)*2916^(n+6).
a(n) = (6*n)!/(5*n+1)!. - Mark van Hoeij, May 29 2013
E.g.f.: exp( 1/6 * Sum_{k>=1} binomial(6*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^5).
a(n) = Sum_{k=0..n} (5*n+1)^(k-1) * |Stirling1(n,k)|. (End)
a(n) ~ 6^(6*n+1/2) * n^(n-1) / (5^(5*n+3/2) * exp(n+1/(5*n))). - Amiram Eldar, Nov 07 2025
MAPLE
spec := [S, {B=Prod(Z, S, S, S, S, S), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# Alternative:
seq((6*n)!/(5*n+1)!, n=0..20); # Mark van Hoeij, May 29 2013
MATHEMATICA
a[n_] := (6*n)!/(5*n+1)!; Array[a, 20, 0] (* Amiram Eldar, Nov 07 2025 *)
PROG
(PARI) a(n) = (6*n)!/(5*n+1)!; \\ Joerg Arndt, May 29 2013
(Python)
from sympy import ff
def A052795(n): return ff(6*n, n-1) # Chai Wah Wu, Sep 01 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
New name using Mark van Hoeij's formula from Joerg Arndt, Feb 18 2019
Accidentally removed a(0) reinserted by Georg Fischer, May 09 2021
STATUS
approved