A003046 -id:A003046 - cp's OEIS frontend

A259711 Numbers that can be written as a product of Catalan numbers.

1, 2, 4, 5, 8, 10, 14, 16, 20, 25, 28, 32, 40, 42, 50, 56, 64, 70, 80, 84, 100, 112, 125, 128, 132, 140, 160, 168, 196, 200, 210, 224, 250, 256, 264, 280, 320, 336, 350, 392, 400, 420, 429, 448, 500, 512, 528, 560, 588, 625, 640, 660, 672, 700, 784, 800, 840

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

			70 = 5*14 = C(3)*C(4). - _David A. Corneth_, Mar 26 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A001013, A003046, A005568, A014228, A259711, A342288.

A203195 (n-1)-st elementary symmetric function of the first n Catalan numbers (A000108).

Original entry on oeis.org

1, 2, 5, 27, 388, 16436, 2175432, 934036488, 1336005150480, 6496133192508960, 109111368146935714560, 6414259771390908431554560, 1334245289372299128934618629120, 991211300949703256415518451506380800

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A000108, A003046 (n-th elem. symm. func.), A014137 (1st elm. symm. func.).

f[k_] := (1/k) Binomial[2 k - 2, k - 1];
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 14}]  (* A203195 *)

A248330 The product of the first n Catalan numbers and the number of standard Young tableaux of shape(1,2,...,n).

Original entry on oeis.org

1, 1, 4, 160, 107520, 1722040320, 854352419880960, 16185399027773630054400, 13931397052191274338996977664000, 632089112919018408339999461491467091968000, 1721041721929360607907210006858724622834371563356160000

Offset: 0

Alejandro H. Morales, Oct 04 2014

The volume of a certain polytope (the Tesler polytope) whose lattice points are Tesler matrices (A008608), and with (n+1)! integral vertices (permutation Tesler matrices).

This is also the iterated constant term of the rational function (x1+x2+...+xn+x(n+1))^binomial(n+1,2)*product_{1<=i

K. Mészáros, A.H. Morales, B. Rhoades, The polytope of Tesler matrices, ArXiv:1409.8566, 2014.

Cf. A000108, A003046, A005118, A008608.

A248330 := proc(n) local i; mul(binomial(2*k, k)/(1+k), k=0..n)*binomial(n+1, 2)!/ mul( (2*i+1)^(n-i), i=0..n-1 ); end;

a(n) = A005118(n+1) * A003046(n).

a(n) = A005118(n+1) * Product_{k=0..n} A000108(k).

A324565 a(n) = Product_{i=0..n, j=0..n} (C(i)+C(j)), where C(k) = Catalan(k) = A000108(k).

Original entry on oeis.org

2, 16, 5184, 3292047360, 431257150704844800000, 1660988174371634812975670031600844800000, 29842277901990912803108442836098281105491323166448779304684748800000

Offset: 0

Vaclav Kotesovec, Mar 07 2019

nonn

Cf. A000108, A003046, A324566.

Table[Product[CatalanNumber[i] + CatalanNumber[j], {i, 0, n}, {j, 0, n}], {n, 0, 7}]

a(n) ~ c * d^n * 2^(4*n^3/3 + 3*n^2) * exp(3*n^2/4) / (Pi^(n^2/2) * n^(3*n^2/2 + 3*n - alfa)), where
d = 1.755243767008515125359629831741161625491476427596217377285488419728083846...
alfa = 0.033913207252709141540468528132950860454628430850651543265936300855299...
c = 1.31959917222402005903928675604752142566298009509...

A247859 The product of the first n Catalan numbers and 2^(n^2).

Original entry on oeis.org

1, 2, 32, 5120, 9175040, 197300060160, 53337309063413760, 187446932178571288903680, 8783433335287216312557974323200, 5597436690584888372318289416604667084800, 49290698636690081763273206158480893991348233830400, 6076713947745931800683801366458443411856602743866957548748800

Offset: 0

Alejandro H. Morales, Sep 25 2014

The volume of a certain polytope (the type D_(n+2) Chan-Robbins-Yuen polytope). This was conjectured by Meszaros-Morales and proved independently by Zeilberger and Kim, both using a variant of the Morris constant term identity (just as for the original Chan-Robbins-Yuen polytope).

J. S. Kim, Proof of a conjecture of Mészáros and Morales on the volume of a flow polytope, arXiv:1407.3467, 2014.
K. Mészáros, A. H. Morales, Flow polytopes of signed graphs and the Kostant partition function, ArXiv:1208.0140, 2012.
D. Zeilberger, Sketch of a Proof of an Intriguing Conjecture of Karola Mészáros and Alejandro Morales Regarding the Volume of the Dn Analog of the Chan-Robbins-Yuen Polytope (Or: The Morris-Selberg Constant Term Identity Strikes Again!), arXiv:1407.2829, 2014.

Cf. A000108 (Catalan numbers).

Cf. A003046 (Product of first n Catalan numbers).

seq(2^(n^2)*mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13);

a[n_] := 2^(n^2)*Product[ CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 13}]

a(n) = 2^(n^2) * A003046(n).

a(n) = 2^(n^2) * prod(k=0..n) A000108(k).

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A259711 Numbers that can be written as a product of Catalan numbers.

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A203195 (n-1)-st elementary symmetric function of the first n Catalan numbers (A000108).

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A248330 The product of the first n Catalan numbers and the number of standard Young tableaux of shape(1,2,...,n).

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A324565 a(n) = Product_{i=0..n, j=0..n} (C(i)+C(j)), where C(k) = Catalan(k) = A000108(k).

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A247859 The product of the first n Catalan numbers and 2^(n^2).

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