A370055 -id:A370055 - 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

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

A370054 a(n) = 2(3n+1)!/(2*n+2)!.

Original entry on oeis.org

1, 2, 14, 180, 3432, 87360, 2790720, 107442720, 4845456000, 250637587200, 14631376032000, 951675588864000, 68257101465907200, 5352223511771136000, 455529588681155788800, 41824228767217408512000, 4120692998969056333824000, 433653882272457833226240000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A001763, A370055.

Cf. A000142, A006013.

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

E.g.f.: exp( 2/3 * Sum_{k>=1} binomial(3*k,k) * x^k/k ).
a(n) = A000142(n)*A006013(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))^2.
a(n) = 2 * Sum_{k=0..n} (2*n+2)^(k-1) * |Stirling1(n,k)|. (End)

A380647 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x)/(1 + x)^3 ).

Original entry on oeis.org

1, 6, 105, 3246, 146637, 8780688, 657224901, 59140486800, 6223651526457, 750357182131200, 102014741343847329, 15443915464974191616, 2576937457466957107845, 469914373917914931984384, 92982800086882512621716925, 19843243096453465663599962112, 4543276116844426827394718716401

Offset: 0

Seiichi Manyama, Feb 06 2025

nonn

Index entries for reversions of series

Cf. A088690, A380646, A380648.
Cf. A370055, A377830.
Cf. A377893.

nmax=17; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-3*x]/(1 + x)^3 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

a(n) = 3*n!*sum(k=0, n, (3*n+3)^(k-1)*binomial(3*n+3, n-k)/k!);

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * exp(3 * x * A(x)).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377893.
a(n) = 3 * n! * Sum_{k=0..n} (3*n+3)^(k-1) * binomial(3*n+3,n-k)/k!.

A370327 E.g.f.: exp(Sum_{k>=1} binomial(3k,k) x^k).

Original entry on oeis.org

1, 3, 39, 801, 22329, 783963, 33142959, 1637118297, 92464016913, 5874794703027, 414582627839319, 32165264232976977, 2720659595107779081, 249128649448324321419, 24549819503580405230751, 2590147977759401893135497, 291286776584800990966021281, 34781064869030286562513208163

Offset: 0

Vaclav Kotesovec, Feb 15 2024

nonn

Cf. A005809, A229451, A370055, A370326.

CoefficientList[Series[Exp[Sum[Binomial[3*k, k]*x^k, {k, 1, 20}]], {x, 0, 20}], x] * Range[0, 20]!
CoefficientList[Series[Exp[2*Cos[ArcCos[1 - 27*x/2]/6] / Sqrt[4 - 27*x] - 1], {x, 0, 20}], x] * Range[0, 20]!

E.g.f.: exp(2*cos(arccos(1 - 27*x/2)/6) / sqrt(4 - 27*x) - 1).

a(n) ~ 3^(3*n - 1/3) * exp((3/2)^(4/3)*n^(1/3) - n - 5/6) * n^(n - 1/3) / 2^(2*n + 1/6).

A375869 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)^(2/3)) - 1) ).

Original entry on oeis.org

1, 3, 24, 327, 6405, 164856, 5276523, 202365351, 9055962270, 463552982301, 26725378964169, 1714193590625478, 121100759112660789, 9344673700445352639, 782093803535217656256, 70570503124491323693523, 6829746633650550406177713, 705729134240394228512985960

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A349598, A375868.

Cf. A027710, A370055.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A349598.

a(n) = 3 * Sum_{k=0..n} (2*n+3)^(k-1) * Stirling2(n,k).

cp's OEIS Frontend

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

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

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A370054 a(n) = 2*(3*n+1)!/(2*n+2)!.

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A380647 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x)/(1 + x)^3 ).

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Mathematica

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A370327 E.g.f.: exp(Sum_{k>=1} binomial(3*k,k) * x^k).

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Mathematica

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A375869 E.g.f. satisfies A(x) = exp( 3 * (exp(x*A(x)^(2/3)) - 1) ).

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Programs

PARI

Formula

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

A370054 a(n) = 2(3n+1)!/(2*n+2)!.

A370327 E.g.f.: exp(Sum_{k>=1} binomial(3k,k) x^k).