A370058 -id:A370058 - cp's OEIS frontend

A065866 a(n) = n! * Catalan(n+1).

1, 2, 10, 84, 1008, 15840, 308880, 7207200, 196035840, 6094932480, 213322636800, 8303173401600, 355850288640000, 16653793508352000, 845180020548864000, 46236318771202560000, 2712530701243883520000, 169890080762116915200000, 11314679378756986552320000

Offset: 0

Len Smiley, Dec 06 2001

nonn

From Noam Zeilberger, Mar 19 2019: (Start)
a(n) is the number of flags in the associahedron of dimension n. For example, there are a(2) = 10 flags in the associahedron of dimension 2, a pentagon. (In this case a flag corresponds to a triple v:e:f of a mutually incident vertex v, edge e, and face f, with f necessarily the unique face of the pentagon.)
Equivalently, a(n) is the number of maximal sequences of consistent parenthesizations of a string of n + 2 letters, starting with n + 1 pairs of parentheses, then removing one pair, and so on, ending with the trivial (outermost) parenthesization. For example, (a(b(cd))):(ab(cd)):(abcd) and (a(b(cd))):(a(bcd)):(abcd) are two of the a(2) = 10 maximal sequences of consistent parenthesizations of the letters abcd. (End)

			G.f. = 1 + 2*x + 10*x^2 + 84*x^3 + 1008*x^4 + 15840*x^5 + 308880*x^6 + ...

R. L. Graham, D. E. Knuth, and O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, pp. 200-204.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000108, A001761.
Cf. A370055, A370058.
Equals 2 * A102693(n+1), n > 0.
Main diagonal of A256116.

List([0..20], n-> 2*Factorial(2*n+1)/Factorial(n+2)); # G. C. Greubel, Mar 19 2019

[Factorial(n)*Catalan(n+1): n in [0..20]]; // G. C. Greubel, Mar 19 2019

with(combstruct): ZL:=[T, {T=Union(Z, Prod(Epsilon, Z, T), Prod(T, Z, Epsilon), Prod(T, T, Z))}, labeled]: seq(count(ZL, size=i+1)/(i+1), i=0..18); # Zerinvary Lajos, Dec 16 2007
a := n -> (2^(2*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*(n+1)*(n+2)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Mar 20 2019

Table[2*(2n+1)!/(n+2)!, {n,0,20}] (* G. C. Greubel, Mar 19 2019 *)
Table[n! CatalanNumber[n+1],{n,0,20}] (* Harvey P. Dale, Feb 02 2023 *)

{ for (n = 0, 100, a = 2 * (2*n + 1)!/(n + 2)!; write("b065866.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

[factorial(n)*catalan_number(n+1) for n in (0..20)] # G. C. Greubel, Mar 19 2019

a(n) = 2 * (2n+1)!/(n+2)!.
E.g.f.: (1-2*x-sqrt(1-4*x))/(2*x^2) = (O.g.f. for A000108)^2 = B_2(x)^2 (cf. GKP reference).
0 = a(n)*(-7200*a(n+2) + 2700*a(n+3) + 246*a(n+4) - 33*a(n+5)) + a(n+1)*(+36*a(n+2) + 372*a(n+3) + 36*a(n+4) - a(n+5)) + a(n+2)*(-18*a(n+2) + 9*a(n+3) + a(n+4)) for n >= 0. - Michael Somos, Apr 14 2015
The e.g.f. A(x) satisfies 0 = -2 + A(x) * (6*x - 2) + A'(x) * (4*x^2 - x). - Michael Somos, Apr 14 2015
(n+2)*a(n) - 2*n*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015
a(n) ~ 4^n*exp(-n)*n^(n - 2)*sqrt(2)*(24*n - 61)/6. - Peter Luschny, Mar 20 2019
Sum_{n>=0} 1/a(n) = (25*exp(1/4)*sqrt(Pi)*erf(1/2) + 22)/32, where erf is the error function. - Amiram Eldar, Dec 04 2022
a(n) = 2 * Sum_{k=0..n} (n+2)^(k-1) * |Stirling1(n,k)|. - Seiichi Manyama, Aug 31 2024
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^2 ). - Seiichi Manyama, Feb 06 2025

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

Original entry on oeis.org

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)

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)

A370055 a(n) = 3(3n+2)!/(2*n+3)!.

Original entry on oeis.org

1, 3, 24, 330, 6552, 171360, 5581440, 218045520, 9945936000, 519177859200, 30535045632000, 1998518736614400, 144098325316915200, 11350405033583616000, 969837188805041356800, 89351761457237190912000, 8830056426362263572480000, 931769828125956695715840000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A001763, A370054.
Cf. A065866, A370058.
Cf. A076151.
Cf. A000142, A001764.

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

E.g.f.: exp( Sum_{k>=1} binomial(3*k,k) * x^k/k ).
a(n) = 3*A076151(n+1) for n > 0.
a(n) = A000142(n)*A001764(n+1). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(2/3))^3.
a(n) = 3 * Sum_{k=0..n} (2*n+3)^(k-1) * |Stirling1(n,k)|. (End)
E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^3 ). - 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)

A375872 E.g.f. satisfies A(x) = exp( 4 * (exp(x*A(x)^(3/4)) - 1) ).

Original entry on oeis.org

1, 4, 44, 836, 22980, 832888, 37615340, 2038258804, 128989219860, 9343113460308, 762596057754748, 69273095355130488, 6932765720797549924, 758009268677055714964, 89907747171907593677068, 11498798927333436173636612, 1577528093912610651931113908

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A349683, A375870, A375871.

Cf. A078944, A370058.

a(n) = 4*sum(k=0, n, (3*n+4)^(k-1)*stirling(n, k, 2));

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A349683.

a(n) = 4 * Sum_{k=0..n} (3*n+4)^(k-1) * Stirling2(n,k).

A380648 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-4*x)/(1 + x)^4 ).

Original entry on oeis.org

1, 8, 188, 7816, 475096, 38289504, 3857806144, 467330651456, 66209818738176, 10747317030192640, 1967261819870112256, 400989528160028255232, 90087157573721153554432, 22119056538323287540637696, 5893098619063477612068864000, 1693364632974231188010697990144

Offset: 0

Seiichi Manyama, Feb 06 2025

nonn

Index entries for reversions of series

Cf. A088690, A380646, A380647.

Cf. A370058.

nmax=16; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-4*x]/(1 + x)^4, {x, 0, nmax}]], x]Range[0, nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

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

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^4 * exp(4 * x * A(x)).

a(n) = 4 * n! * Sum_{k=0..n} (4*n+4)^(k-1) * binomial(4*n+4,n-k)/k!.

cp's OEIS Frontend

A065866 a(n) = n! * Catalan(n+1).

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A365340 a(n) = (4*n)!/(3*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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A370055 a(n) = 3*(3*n+2)!/(2*n+3)!.

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

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A375872 E.g.f. satisfies A(x) = exp( 4 * (exp(x*A(x)^(3/4)) - 1) ).

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Formula

A380648 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-4*x)/(1 + x)^4 ).

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Programs

Mathematica

PARI

Formula

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

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

A370055 a(n) = 3(3n+2)!/(2*n+3)!.

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