A365340 -id:A365340 - cp's OEIS frontend

A365341 a(n) = (5n)!/(4n+1)!.

1, 1, 10, 210, 6840, 303600, 17100720, 1168675200, 93963542400, 8691104822400, 909171781056000, 106137499051584000, 13679492361575040000, 1929327666754295808000, 295570742023171270656000, 48877281133334949335040000, 8677556868736487617966080000

Offset: 0

Seiichi Manyama, Sep 01 2023

Cf. A001761, A001763, A052795, A365340.
Cf. A004343.
Cf. A000142, A002294.

PARI
```
a(n) = (5*n)!/(4*n+1)!;
```

from sympy import ff
def A365341(n): return ff(5*n,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: exp( 1/5 * Sum_{k>=1} binomial(5*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002294(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)^4).
a(n) = Sum_{k=0..n} (4*n+1)^(k-1) * |Stirling1(n,k)|. (End)

A370057 a(n) = 3(4n+2)!/(3*n+3)!.

Original entry on oeis.org

1, 3, 30, 546, 14688, 526680, 23680800, 1282554000, 81339793920, 5915366392320, 485415660038400, 44376781223174400, 4473125162795520000, 492902545595556096000, 58949616073242166272000, 7605168496387089788160000, 1052810955815818170875904000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..327

Cf. A365340, A370056, A370058.

Cf. A000142, A006632.

Table[(3(4n+2)!)/(3n+3)!,{n,0,20}] (* Harvey P. Dale, Feb 15 2025 *)

PARI
```
a(n) = 3*(4*n+2)!/(3*n+3)!;
```

E.g.f.: exp( 3/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A006632(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x))^3.
a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * |Stirling1(n,k)|. (End)

A370058 a(n) = 4(4n+3)!/(3*n+4)!.

Original entry on oeis.org

1, 4, 44, 840, 23256, 850080, 38750400, 2120489280, 135566323200, 9922550077440, 818544054182400, 75160674504115200, 7604312776752384000, 840608992488545280000, 100812386907863414784000, 13037431708092153922560000, 1808675231786149165350912000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A365340, A370056, A370057.
Cf. A065866, A370055.
Cf. A000142, A002293.

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

E.g.f.: exp( Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A002293(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+4)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(3/4))^4.
a(n) = 4 * Sum_{k=0..n} (3*n+4)^(k-1) * |Stirling1(n,k)|. (End)
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^4 ). - Seiichi Manyama, Feb 06 2025

A370056 a(n) = 2(4n+1)!/(3*n+2)!.

Original entry on oeis.org

1, 2, 18, 312, 8160, 287280, 12751200, 684028800, 43062243840, 3113350732800, 254265345734400, 23153103246873600, 2326025084653670400, 255579097716214272000, 30491180727539051520000, 3925248256199788277760000, 542357159056633603178496000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A365340, A370057, A370058.

Cf. A000142, A069271.

PARI
```
a(n) = 2*(4*n+1)!/(3*n+2)!;
```

E.g.f.: exp( 1/2 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A069271(n).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*a(n) -8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(3/2))^2.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * |Stirling1(n,k)|. (End)

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

A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 11, 160, 3941, 134486, 5851327, 309520436, 19283504585, 1382980764106, 112223497464371, 10165461405056552, 1016801830348902061, 111312715288354681310, 13237965546409421546471, 1699516550894276788156156, 234263144339070269872076177, 34507561203827621878485498386

Offset: 0

Seiichi Manyama, Mar 12 2025

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 1, 0, 1, 1, 2, 1, 0, 1, ...] with period 5. In particular, a(5*n+3) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A381984, A381988, A381989.

Cf. A002293, A365340.

seq(simplify(hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27)), n = 0..17); # Peter Bala, Mar 13 2025

Table[HypergeometricPFQ[{-n, 1/2, 1/4, 3/4}, {2/3, 4/3}, -256/27], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)

a(n) = n!*sum(k=0, n, binomial(4*k+1, k)/((4*k+1)*(n-k)!));

a(n) = n! * Sum_{k=0..n} A002293(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27).
3*(3*n - 1)*(3*n + 1)*a(n) = n*(256*n^2 - 303*n + 95)*a(n-1) - 3*(n - 1)*(256*n^2 - 485*n + 245)*a(n-2) + 3*(256*n - 375)*(n - 1)*(n - 2)*a(n-3) - 256*(n - 1)*(n - 2)*(n - 3)*a(n-4) with a(0) = 1, a(1) = 2, a(2) = 11 and a(3) = 160. (End)
a(n) ~ 2^(8*n+1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 27/256)). - Vaclav Kotesovec, Mar 14 2025

cp's OEIS Frontend

A365341 a(n) = (5*n)!/(4*n+1)!.

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A370057 a(n) = 3*(4*n+2)!/(3*n+3)!.

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Mathematica

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A370058 a(n) = 4*(4*n+3)!/(3*n+4)!.

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A370056 a(n) = 2*(4*n+1)!/(3*n+2)!.

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

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Maple

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A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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Author

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Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A365341 a(n) = (5n)!/(4n+1)!.

A370057 a(n) = 3(4n+2)!/(3*n+3)!.

A370058 a(n) = 4(4n+3)!/(3*n+4)!.

A370056 a(n) = 2(4n+1)!/(3*n+2)!.

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