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 1-4 of 4 results.

A004319 a(n) = binomial(3*n, n - 1).

Original entry on oeis.org

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

Crossrefs

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).
Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

Programs

  • Maple
    A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015
  • Mathematica
    Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)
  • Maxima
    a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */
  • PARI
    vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

Formula

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005
D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013
G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015
a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016
a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024

A004331 Binomial coefficient C(4n,n-1).

Original entry on oeis.org

1, 8, 66, 560, 4845, 42504, 376740, 3365856, 30260340, 273438880, 2481256778, 22595200368, 206379406870, 1889912732400, 17345898649800, 159518999862720, 1469568786235308, 13559593014190944, 125288932441604200
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

Crossrefs

Cf. A002293, A005810, A052203, A224274, A257633, A262977.
Cf. A001791, A004319, A004343, A004356, A004369, A004382.

Programs

  • Maple
    #A004331
seq(binomial(4*n - 1,n), n = 0..20);
  • Mathematica
    a[n_] := Binomial[4*n, n - 1]; Array[a, 19] (* Amiram Eldar, May 09 2020 *)
  • PARI
    vector(30, n, binomial(4*n, n-1)) \\ Altug Alkan, Nov 05 2015

Formula

G.f.: (g^2-g)/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
With an offset of 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A224274 (k = 3). - Peter Bala, Nov 04 2015
D-finite with recurrence 3*(n-1)*(3*n-1)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

A004343 Binomial coefficient C(5n,n-1).

Original entry on oeis.org

1, 10, 105, 1140, 12650, 142506, 1623160, 18643560, 215553195, 2505433700, 29248649430, 342700125300, 4027810484880, 47465835030320, 560658857389200, 6635869816740560, 78682166288559225, 934433788613079150
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

Crossrefs

Cf. A001791, A004319, A004331, A004356, A004369, A004382.

Formula

From Peter Bala, Jul 21 2024: (Start)
a(n) = Sum_{k = 0..n-1} binomial(4*n+k, k).
a(n) = 5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/((4*n - 4)*(4*n - 2)*(4*n - 1)*(4*n + 1)) * a(n-1) with a(1) = 1. (End)

A004356 Binomial coefficient C(6n,n-1).

Original entry on oeis.org

1, 12, 153, 2024, 27405, 376992, 5245786, 73629072, 1040465790, 14783142660, 210980549208, 3022285436352, 43430966148115, 625806790696080, 9038619861406740, 130815226545884704, 1896724514424115530
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

Crossrefs

Cf. A001791, A004319, A004331, A004343, A004369, A004382.

Programs

  • Mathematica
    Table[Binomial[6n,n-1],{n,20}] (* Harvey P. Dale, Mar 04 2017 *)

Formula

D-finite with recurrence 5*(n-1)*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 19 2025
Showing 1-4 of 4 results.

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

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