cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A144186 Numerators of series expansion of the e.g.f. for the Catalan numbers.

Original entry on oeis.org

1, 1, 1, 5, 7, 7, 11, 143, 143, 2431, 4199, 4199, 7429, 7429, 7429, 215441, 392863, 392863, 20677, 765049, 765049, 31367009, 58642669, 58642669, 2756205443, 2756205443, 2756205443, 146078888479, 5037203051, 5037203051, 9586934839
Offset: 0

Views

Author

Eric W. Weisstein, Sep 13 2008

Keywords

Examples

			E.g.f. = 1 + x + x^2 + (5*x^3)/6 + (7*x^4)/12 + ...
The coefficients continue like this: 1, 1, 1, 5/6, 7/12, 7/20, 11/60, 143/1680, 143/4032, 2431/181440, 4199/907200, 4199/2851200, 7429/17107200, 7429/62270208, ...

Links

Crossrefs

Cf. A000108, A144187.

Programs

  • Magma
    [Numerator(Binomial(2*n,n)/Factorial(n+1)): n in [0..30]]; // G. C. Greubel, Jan 17 2019
  • Maple
    seq(numer(binomial(2*n,n)/(n+1)!),n=0..30); # Vladeta Jovovic, Dec 03 2008
  • Mathematica
    With[{m = 30}, CoefficientList[Series[E^(2*x)*(BesselI[0, 2*x] - BesselI[1, 2*x]), {x, 0, m}], x]]//Numerator (* G. C. Greubel, Jan 17 2019 *)
  • PARI
    vector(30, n, n--; numerator(binomial(2*n,n)/(n+1)!)) \\ G. C. Greubel, Jan 17 2019
  • Sage
    [numerator(binomial(2*n,n)/factorial(n+1)) for n in (0..30)] # G. C. Greubel, Jan 17 2019

Formula

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

Views

Author

N. J. A. Sloane, Dec 31 2010

Keywords

References

  • 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.

Crossrefs

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

Formula

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

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

Views

Author

N. J. A. Sloane, Dec 31 2010

Keywords

Comments

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

Examples

			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, ...

Crossrefs

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

Views

Author

N. J. A. Sloane, Dec 31 2010

Keywords

Comments

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

Crossrefs

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

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

Views

Author

Sela Fried, Sep 19 2021

Keywords

Comments

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.

Examples

			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, ...
  ...

Links

Crossrefs

Cf. A000108, A000142, A144186, A144187.

Programs

  • Mathematica
    Join@@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,6}] (* Giorgos Kalogeropoulos, Nov 16 2021 *)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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