A365341 -id:A365341 - cp's OEIS frontend

A365340 a(n) = (4n)!/(3n+1)!.

1, 1, 8, 132, 3360, 116280, 5100480, 271252800, 16963914240, 1220096908800, 99225500774400, 9003984596006400, 901928094049382400, 98856066097780992000, 11768525894839633920000, 1512185803617951221760000, 208598907329474462760960000

Offset: 0

Seiichi Manyama, Sep 01 2023

Cf. A001761, A001763, A052795, A365341.
Cf. A004331, A061924.
Cf. A370056, A370057, A370058.
Cf. A000142, A002293.

PARI
```
a(n) = (4*n)!/(3*n+1)!;
```

from sympy import ff
def A365340(n): return ff(n<<2,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002293(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)^3).
a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * |Stirling1(n,k)|. (End)

A365368 First integer > n reached under iteration of map x -> (5/3)*round(x) when started at n, or -1 if no such integer is ever reached.

Original entry on oeis.org

5, 5, 5, 20, 10245, 10, 20, 10245, 15, 130, 30, 20, 10245, 105, 25, 45, 130, 30, 245, 55, 35, 10245, 105, 40, 70, 120, 45, 130, 80, 50, 145, 245, 55, 95, 270, 60, 10245, 105, 65, 520, 870, 70, 120, 2605, 75, 355, 130, 80, 380, 230, 85, 145, 245, 90, 255, 155, 95

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Conjecture: an integer will always be reached, i.e. a(n) > 0 for all n.

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367.

from fractions import Fraction
def A365368(n):
    x = Fraction(n)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._round_(),3)
    return int(x)

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

Original entry on oeis.org

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

Offset: 0

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

Old name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..310
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 752

Cf. A001761, A001763, A365340, A365341.

Cf. A000142, A002295.

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

a(n) = (6*n)!/(5*n+1)!; \\ Joerg Arndt, May 29 2013

from sympy import ff
def A052795(n): return ff(6*n,n-1) # Chai Wah Wu, Sep 01 2023

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)

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

A365369 A365368/5, except when A365368(n) = -1, then a(n) = -1.

Original entry on oeis.org

1, 1, 1, 4, 2049, 2, 4, 2049, 3, 26, 6, 4, 2049, 21, 5, 9, 26, 6, 49, 11, 7, 2049, 21, 8, 14, 24, 9, 26, 16, 10, 29, 49, 11, 19, 54, 12, 2049, 21, 13, 104, 174, 14, 24, 521, 15, 71, 26, 16, 76, 46, 17, 29, 49, 18, 51, 31, 19, 54, 151, 20, 34, 2049, 21, 99, 36

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368.

from fractions import Fraction
def A365369(n):
    x = Fraction(n)
    while x.denominator > 1 or x<=n:
        x = Fraction(5*x._round_(),3)
    return int(x)//5

A365370 Positions of records in A365367.

Original entry on oeis.org

1, 5, 415, 635, 15935, 60971, 275039, 514661, 2857994, 14179544, 170794880, 2382918520

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Numbers k such that iteration of the map x -> (5/3)*round(x) starting at x = k takes more steps to reach an integer > k than it does for any number from 1 to k - 1.

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368.

A365367(a(n)) = A365371(n).

A365371 Record values of A365367.

Original entry on oeis.org

3, 15, 17, 21, 24, 28, 32, 35, 37, 45, 50, 55

Offset: 1

Chai Wah Wu, Sep 02 2023

nonn

Numbers v = A365367(k) such that A365367(m) < v for 1 <= m < k.

Cf. A087704, A087705, A087706, A087707, A087708, A087709, A365341, A365342, A365367, A365368, A365369, A365370.

a(n) = A365367(A365370(n)).

cp's OEIS Frontend

A365340 a(n) = (4*n)!/(3*n+1)!.

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A365368 First integer > n reached under iteration of map x -> (5/3)*round(x) when started at n, or -1 if no such integer is ever reached.

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

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A365369 A365368/5, except when A365368(n) = -1, then a(n) = -1.

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A365370 Positions of records in A365367.

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A365371 Record values of A365367.

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Formula

A365340 a(n) = (4n)!/(3n+1)!.

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