A144186 -id:A144186 - cp's OEIS frontend

A144187 Denominators of series expansion of the e.g.f. for the Catalan numbers.

1, 1, 1, 6, 12, 20, 60, 1680, 4032, 181440, 907200, 2851200, 17107200, 62270208, 242161920, 29059430400, 232475443200, 1077840691200, 277159034880, 52660216627200, 283555012608000, 65501207912448000, 720513287036928000

Offset: 0

Eric W. Weisstein, Sep 13 2008

			E.g.f. = 1 + x + x^2 + (5*x^3)/6 + (7*x^4)/12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..480
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A144186.

[Denominator(Binomial(2*n,n)/Factorial(n+1)): n in [0..30]]; // G. C. Greubel, Jan 17 2019

seq(denom(binomial(2*n,n)/(n+1)!),n=0..30); # Vladeta Jovovic, Dec 03 2008

With[{m = 30}, CoefficientList[Series[Exp[2*x]*(BesselI[0, 2*x] - BesselI[1, 2*x]), {x, 0, m}], x]]//Denominator (* G. C. Greubel, Jan 17 2019 *)

vector(30, n, n--; denominator(binomial(2*n,n)/(n+1)!)) \\ G. C. Greubel, Jan 17 2019

[denominator(binomial(2*n,n)/factorial(n+1)) for n in (0..30)] # G. C. Greubel, Jan 17 2019

E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)).

A178955 E.g.f. is inverse of e.g.f. for Catalan numbers.

Original entry on oeis.org

1, -1, 0, 1, 2, -2, -28, -65, 338, 3262, 4352, -113082, -879140, 1145012, 68641120, 409571279, -3075414734, -67796919090, -235926569056, 6635196777226, 98653814115636, -51631812077716, -17882630766156440, -190179698567684014, 1532579370407751292, 62028205219536446948, 405883930741148425152, -11224575706163698420700, -269584771812788695251352, -338220005828087037972744

Offset: 0

N. J. A. Sloane, Dec 31 2010

sign

Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/, see p. 130.

See A178956/A178957 for the coefficients in the e.g.f. Cf. A000108, A144186/A144187.

The e.g.f. is 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

A157625 Product of the composite numbers between n+1 and 2n, both inclusive.

Original entry on oeis.org

1, 4, 24, 48, 4320, 8640, 120960, 3628800, 7257600, 14515200, 6706022400, 13412044800, 8717829120000, 470762772480000, 941525544960000, 1883051089920000, 2112783322890240000, 147894832602316800000

Offset: 1

Jaume Oliver Lafont, Mar 03 2009

nonn

This function is very useful in a problem due to Paul Erdős recorded in A157017. - M. F. Hasler, Feb 26 2014

Cf. A073840, A157017, A144186 (product of primes between n+2 and 2n, both inclusive).

nn=20;With[{comps=Complement[Range[2nn],Prime[Range[PrimePi[2nn]]]]}, Table[ Times@@ Select[comps,#>n&&#<=2n&],{n,nn}]] (* Harvey P. Dale, Feb 18 2013 *)

PARI
```
a(n)=prod(i=n+1,2*n,if(isprime(i),1,i))
```

a(n) = n!*A000984(n)*A034386(n)/A034386(2n). - M. F. Hasler, Feb 26 2014

A178956 Numerators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

Original entry on oeis.org

1, -1, 0, 1, 1, -1, -7, -13, 169, 233, 17, -18847, -43957, 26023, 429007, 31505483, -31381783, -753299101, -7372705283, 195152846389, 632396244331, -52258919107, -447065769153911, -13584264183406001, 34831349327448893, 674219621951483119, 2113978805943481381, -6602691591860999071, -370308752490094361609, -95867348590727618473, 752151788353653780085507

Offset: 0

N. J. A. Sloane, Dec 31 2010

This is the reciprocal of the e.g.f. for the Catalan numbers.

			1, -1, 0, 1/6, 1/12, -1/60, -7/180, -13/1008, 169/20160, 233/25920, 17/14175, -18847/6652800, -43957/23950080, 26023/141523200, 429007/544864320, 31505483/100590336000, ...

Cf. A000108, A178957 (denominators), A178955, A144186/A144187.

A178957 Denominators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

Original entry on oeis.org

1, 1, 1, 6, 12, 60, 180, 1008, 20160, 25920, 14175, 6652800, 23950080, 141523200, 544864320, 100590336000, 213497856000, 3952082534400, 200074178304000, 3577797070848000, 15595525693440000, 51711479930880000, 28100018194440192000, 1846572624206069760000, 14101100039391805440000, 168600109166641152000000, 2100476360034404352000000, 6405217323775501271040000, 418802671169936621568000000, 2506168365572477878272000000, 2368329105465991594967040000000

Offset: 0

N. J. A. Sloane, Dec 31 2010

nonn

This is the reciprocal of the e.g.f. for the Catalan numbers.

Cf. A000108, A178956 (numerators), A178955, A144186/A144187.

A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 6, 3, 1, 1, 30, 15, 5, 5, 1, 30, 15, 5, 5, 1, 1, 210, 105, 35, 35, 7, 7, 1, 210, 105, 35, 35, 7, 7, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 1, 2310, 1155, 385, 385, 77, 77, 11, 11, 11, 11, 1, 2310, 1155, 385, 385

Offset: 1

Arkadiusz Wesolowski, Nov 27 2011

Row sums give A201156.
Central terms give A068111: T(2*n-1,n) = A068111(n).
T(n,1) = A034386(n).
T(n,n-1) = A089026(n) for n > 1.
T(n,n) = A000012(n).
Let n > 1 and p = A000040(n). Then T(p,p-1) = T(p+1,p-1) = p.
T(2*n-1,n-1) = A073838(n) for n > 1.
T(2*n,n+1) = A144186(n).

			1:                                   1
2:                               2       1
3:                           6       3       1
4:                       6       3       1       1
5:                   30      15      5       5       1
6:               30      15      5       5       1       1
7:           210     105     35      35      7       7       1
8:       210     105     35      35      7       7       1       1
9:   210     105     35      35      7       7       1       1       1

Arkadiusz Wesolowski, Rows n = 1..141 of triangle, flattened

Cf. A034386.

lst = {}; Do[AppendTo[lst, Product[Prime[i], {i, PrimePi[n]}]/Product[Prime[i], {i, PrimePi[k]}]], {n, 12}, {k, n}]; lst (* Arkadiusz Wesolowski, Dec 02 2011 *)

A347917 The coefficients in the expansion x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n), given row by row.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 3, 4, 2, 1, 1, 1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 6, 9, 6, 3, 3, 4, 2, 1, 1, 4, 9, 6, 3, 6, 8, 4, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 3, 4, 2, 1, 1, 1, 2, 1, 1, 1, 1, 5, 4, 3, 2, 1, 10, 16, 12, 8, 4, 6, 9, 6, 3, 3, 4, 2, 1, 1, 10, 24, 18, 12, 6, 18, 27, 18, 9, 9, 12, 6, 3, 3, 4, 9, 6, 3, 6, 8

Offset: 0

Sela Fried, Sep 19 2021

The coefficients are ordered lexicographically and by decreasing degree.
Each row of the triangle consists of C_n numbers where C_n is the n-th Catalan number.
The sum of each row is n!.
In the triangle, the (n+1)-th row contains (at least) two copies of the n-th row.
The average of each row is n!/C_n.

			The fourth row of the triangle is 1,2,1,1,1 since x_1(x_1 + x_2)(x_1 + x_2 + x_3) = x_1^3 + 2x_1^2x_2+x_1x_2^2 + x_1^2x_3+x_1x_2x_3.
The first six rows of the triangle are:
  1
  1
  1, 1
  1, 2, 1, 1, 1
  1, 3, 2, 1, 3, 4, 2, 1, 1, 1, 2, 1, 1, 1
  1, 4, 3, 2, 1, 6, 9, 6, 3, 3, 4, 2, 1, 1, 4, 9, 6, 3, 6, ...
  ...

Sela Fried, On the coefficients of the distinct monomials in the expansion of x1(x1+x2)...(x1+x2+...+xn), arXiv:2111.07331 [math.CO], 2021.
R. P. Stanley, Catalan Addendum, 2008.

Cf. A000108, A000142, A144186, A144187.

Join@@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,6}] (* Giorgos Kalogeropoulos, Nov 16 2021 *)

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A144187 Denominators of series expansion of the e.g.f. for the Catalan numbers.

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A178955 E.g.f. is inverse of e.g.f. for Catalan numbers.

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A157625 Product of the composite numbers between n+1 and 2n, both inclusive.

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A178956 Numerators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

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A178957 Denominators in expansion of 1/(Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1)).

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A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

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A347917 The coefficients in the expansion x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n), given row by row.

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