A052795 a(n) = (6*n)!/(5*n+1)!.

1, 1, 12, 306, 12144, 657720, 45239040, 3776965920, 371090522880, 41951580652800, 5364506808460800, 765606216965990400, 120639963305775513600, 20803502274492921984000, 3896911902445736638464000, 787971434323820421362688000, 171063718698166603304067072000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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

Crossrefs

Cf. A001761, A001763, A365340, A365341.

Cf. A000142, A002295.

Programs

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

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

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)

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