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A052795
a(n) = (6*n)!/(5*n+1)!.
3
1, 1, 12, 306, 12144, 657720, 45239040, 3776965920, 371090522880, 41951580652800, 5364506808460800, 765606216965990400, 120639963305775513600, 20803502274492921984000, 3896911902445736638464000, 787971434323820421362688000, 171063718698166603304067072000
OFFSET
0,3
COMMENTS
Old name was: A simple grammar.
LINKS
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}.
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
a(n) = A000142(n)*A002295(n). - Alois P. Heinz, 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)
MAPLE
spec := [S, {B=Prod(Z, S, S, S, S, S), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); # end of program
seq((6*n)!/(5*n+1)!, n=0..20); # Mark van Hoeij, May 29 2013
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
encyclopedia(AT)pommard.inria.fr, 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