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 1-3 of 3 results.

A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).

Original entry on oeis.org

1, -6, -9, -36, -189, -1134, -7371, -50544, -360126, -2640924, -19806930, -151252920, -1172210130, -9197341020, -72921775230, -583374201840, -4703454502335, -38180983607190, -311811366125385, -2560135427134740, -21121117273861605, -175003543126281870, -1455711290550435555
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A004988, A004990, A004991, A004992, A025748, A155579, A185047.

Programs

  • GAP
    List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k-2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
  • Magma
    [1] cat [3^n*(&*[3*k-2: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
  • Maple
    a:= n-> (3^n/n!)*product(3*k-2, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019
  • Mathematica
    Table[9^n*Pochhammer[-2/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
  • PARI
    a(n)=if(n<0,0,prod(k=0,n-1,3*k-2)*3^n/n!)
  • Sage
    [9^n*rising_factorial(-2/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

Formula

a(n) ~ -(2/3)*Gamma(1/3)^-1*n^(-5/3)*3^(2*n)*(1 + (5/9)*n^-1 + ...).
G.f.: (1-9*x)^(2/3).
D-finite with recurrence: n*a(n) +3*(-3*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 99/128 - 5*sqrt(3)*Pi/512 - 15*log(3)/512. - Amiram Eldar, Dec 02 2022
From Peter Bala, Oct 14 2024: (Start)
a(n) = -1/(sqrt(3)*Pi) * 9^n * Gamma(2/3)*Gamma(n-2/3)/Gamma(n+1).
For n >= 1, a(n) = - Integral_{x = 0..9} x^n * w(x) dx, where w(x) = sqrt(3)/(2*Pi) * x^(1/3)*(9 - x)^(2/3)/x^2. (End)

A034164 Related to triple factorial numbers 2*A034000(n+1).

Original entry on oeis.org

1, 5, 30, 198, 1386, 10098, 75735, 580635, 4528953, 35819901, 286559208, 2314516680, 18846778680, 154543585176, 1274984577702, 10574872085646, 88123934047050, 737458184920050, 6194648753328420, 52212039492339540, 441429061162507020, 3742550735942994300
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A004990, A025748, A034000, A073006, A185047.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // G. C. Greubel, Sep 17 2019
  • Maple
    seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # G. C. Greubel, Sep 17 2019
  • Mathematica
    CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)
  • PARI
    my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ G. C. Greubel, Sep 17 2019
  • Sage
    def A034164_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
A034164_list(30) # G. C. Greubel, Sep 17 2019

Formula

a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
G.f.: 2F1( (1, 5/3); 3; 9 x ). - Olivier Gérard, Feb 15 2011
D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson, Nov 07 2015
Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - Amiram Eldar, Dec 02 2022
a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - Amiram Eldar, Aug 19 2025

A155579 Recursive sequence (n+1)*a(n) = 3*(3*n-2)*a(n-1).

Original entry on oeis.org

2, 3, 12, 63, 378, 2457, 16848, 120042, 880308, 6602310, 50417640, 390736710, 3065780340, 24307258410, 194458067280, 1567818167445, 12726994535730, 103937122041795, 853378475711580, 7040372424620535, 58334514375427290
Offset: 0

Views

Author

Roger L. Bagula, Jan 24 2009

Keywords

Comments

This is built akin to (n+1)*C(n) = 2*(2*n-1)*C(n-1) for the Catalan numbers A000108.

Links

Crossrefs

Cf. A185047, A283150.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = ((3*n - 2)/(n + 1))*a[n - 1];
Table[2*3^(n)*a[n], {n, 0, 30}]
  • Maxima
    a(n):=3^(2*n-1)*sum(binomial(k,n-k)*2^(2*k-n)*(-1)^(n-k)*(if k=1 then (1/3) else 1/k*(1/3)^k*sum(binomial(i,k-1-i)*(-1/3)^(k-1-i)*binomial(i+k-1,k-1),i,1,k-1)),k,1,n); /* Vladimir Kruchinin, Sep 20 2010 */

Formula

(n+1)*a(n) = 3*(3*n-2)*a(n-1).
From Vladimir Kruchinin, Sep 20 2010: (Start)
G.f.: A(x) = 1/3*(1-(1-9*x)^(2/3)).
a(n) = 3^(2*n-1)*sum(binomial(k,n-k)*2^(2*k-n)*(-1)^(n-k)*(if k=1 then (1/3) else 1/k*(1/3)^k*sum(binomial(i,k-1-i)*(-1/3)^(k-1-i)*binomial(i+k-1,k-1),i,1,k-1)),k,1,n),n>0. (End)
From Vaclav Kotesovec, Jul 20 2019: (Start)
a(n) = 2 * 3^(2*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n+2)).
a(n) ~ 2 * 3^(2*n) / (Gamma(1/3) * n^(5/3)). (End)
a(n) = 3*A185047(n-1) for n >= 1. - Peter Bala, Oct 14 2024
Showing 1-3 of 3 results.

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