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.

A004988 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 2).

Original entry on oeis.org

1, 6, 45, 360, 2970, 24948, 212058, 1817640, 15677145, 135868590, 1182056733, 10316131488, 90266150520, 791564704560, 6954461332920, 61199259729696, 539318476367946, 4758692438540700, 42035116540442850, 371678925199705200, 3289358488017391020
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A004987, A004989, A004990, A004991, A004992, A073006.

Programs

  • GAP
    List([0..20], 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..20]]; // G. C. Greubel, Aug 22 2019
  • Maple
    A004988 := proc(n)
        binomial(-2/3,n)*(-9)^n ;
end proc: # R. J. Mathar, Sep 16 2012
  • Mathematica
    Table[FullSimplify[9^n*Gamma[n+2/3]/(Gamma[2/3]*Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-9x)^(-2/3), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 10 2014 *)
Table[9^n*Pochhammer[2/3, n]/n!, {n,0,20}] (* 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..20)] # G. C. Greubel, Aug 22 2019

Formula

G.f.: (1-9*x)^(-2/3).
a(n) = 9^n*Gegenbauer_C(n,1/3,1). - Paul Barry, Apr 21 2009
a(n) = Product_{k=1..n} (9 - 3/k). - Michel Lagneau, Sep 16 2012
a(n) = (-9)^n*binomial(-2/3, n). - R. J. Mathar, Sep 16 2012
D-finite with recurrence: n*a(n) +3*(-3*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
a(n) = 9^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 9/8 + sqrt(3)*Pi/32 - 3*log(3)/32. - Amiram Eldar, Dec 02 2022
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) = sqrt(3)/(2*Pi) * 1/(x*(9 - x)^2)^(1/3). The weight function w(x) is the solution of the Hausdorff moment problem on (0, 9) with moments equal to a(n). As a consequence this representation is unique. Cf. A004987. - Peter Bala, Oct 13 2024
a(n) ~ c * 9^n / n^(1/3), where c = 1/Gamma(2/3) = 1/A073006 = 0.738488... . - Amiram Eldar, Aug 17 2025

A004990 a(n) = (3^n/n!)*Product_{k=0..n-1} (3*k - 1).

Original entry on oeis.org

1, -3, -9, -45, -270, -1782, -12474, -90882, -681615, -5225715, -40760577, -322379109, -2579032872, -20830650120, -169621008120, -1390892266584, -11474861199318, -95173848770814, -793115406423450, -6637123664280450, -55751838779955780
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A004988, A004989, A004991, A004992, A034164.

Programs

  • GAP
    List([0..20], n-> 3^n*Product([0..n-1], k-> 3*k-1)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
  • Magma
    [1] cat [3^n*(&*[3*k-1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 22 2019
  • Maple
    a:= n-> (3^n/n!)*mul(3*k-1, k=0..n-1): seq(a(n), n=0..20); # G. C. Greubel, Aug 22 2019
  • Mathematica
    FullSimplify[Table[3^(2*n) * Gamma[n-1/3] / (n! * Gamma[-1/3]),{n,0,20}]] (* Vaclav Kotesovec, Dec 03 2014 *)
  • PARI
    for(n=0,30,print1( (3^n/n!)*prod(k=0,n-1,(3*k-1) ),","))
  • Sage
    [9^n*rising_factorial(-1/3, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 22 2019

Formula

a(n) = 9*A034164(n-2), n >= 2.
G.f.: (1 - 9*x)^(1/3).
a(n) ~ -1/3*Gamma(2/3)^-1*n^(-4/3)*3^(2*n)*{1 + 2/9*n^-1 + ...}.
G.f.: 1 + 3*x/(G(0)-3*x) where G(k) = (1+9*x)*k + 1 - 3*x - 3*x*(k+1)*(3*k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 07 2012
D-finite with recurrence: n*a(n) +3*(-3*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 9/16 - sqrt(3)*Pi/64 + 3*log(3)/64. - Amiram Eldar, Dec 02 2022
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-9)^n*binomial(1/3, n).
E.g.f.: hypergeom([-1/3], [1], 9*x).
a(n) = (9^n)*Sum_{k = 0..2*n} (-1)^k*binomial(1/3, k)* binomial(1/3, 2*n - k).
(9^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = A004989.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = 18^n/(2*n)! * Product_{k = 1..n} (6*k - 5)*(3*k - 4). (End)

Extensions

More terms from Jason Earls, Dec 03 2001

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)

A004992 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k + 5).

Original entry on oeis.org

1, 15, 180, 1980, 20790, 212058, 2120580, 20902860, 203802885, 1970094555, 18912907728, 180532301040, 1715056859880, 16227076443480, 152998149324240, 1438182603647856, 13482961909198650, 126105349621328550, 1176983263132399800, 10964528293391303400, 101970113128539121620
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A004988, A004989, A004990, A004991.

Programs

  • GAP
    List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k+5)/Factorial(n) ); # G. C. Greubel, Aug 22 2019
  • Magma
    [1] cat [3^n*&*[3*k+5: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
  • Maple
    a:= n-> (3^n/n!)*mul(3*k+5, k=0..n-1): seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019
  • Mathematica
    Table[9^n*Pochhammer[5/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)
Table[3^n/n! Product[3k+5,{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Jan 07 2023 *)
  • PARI
    a(n) = 3^n*prod(k=0,n-1, 3*k+5)/n!;
vector(25, n, a(n-1)) \\ G. C. Greubel, Aug 22 2019
  • Sage
    [9^n*rising_factorial(5/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

Formula

G.f.: (1 - 9*x)^(-5/3).
a(n) ~ (3/2)*Gamma(2/3)^-1*n^(2/3)*3^(2*n)*(1 + (5/9)*n^-1 - ...).
D-finite with recurrence: n*a(n) +3*(-3*n-2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/2 - 3*log(3)/2. - Amiram Eldar, Dec 02 2022

Extensions

Terms a(16) onward added by G. C. Greubel, Aug 22 2019

A382516 Expansion of 1/(1 - x/(1 - 9*x)^(4/3)).

Original entry on oeis.org

1, 1, 13, 151, 1693, 18688, 204631, 2230498, 24246229, 263112874, 2852058448, 30892668295, 334454025715, 3619669508056, 39164977065622, 423695451762664, 4583082589819489, 49570596449054509, 536121822834121354, 5798064369702626227, 62702959640721355228
Offset: 0

Views

Author

Seiichi Manyama, Mar 30 2025

Keywords

Crossrefs

Cf. A362206, A362210, A382517.
Cf. A004991.

Programs

  • PARI
    a(n) = sum(k=0, n, 9^(n-k)*binomial(n+k/3-1, n-k));

Formula

a(n) = Sum_{k=0..n} 9^(n-k) * binomial(n+k/3-1,n-k).
D-finite with recurrence (n-1)*(n-2)*a(n) -3*(n-2)*(17*n-35)*a(n-1) +27*(39*n^2-197*n+252)*a(n-2) +2*(-5468*n^2+32199*n-46873)*a(n-3) +6*(9115*n^2-56514*n+77702)*a(n-4) +54*(-1094*n^2-359*n+28901)*a(n-5) +54*(-9846*n^2+134559*n-449254)*a(n-6) +177147*(3*n-19)*(3*n-20)*a(n-7)=0. - R. J. Mathar, Mar 31 2025
Showing 1-5 of 5 results.

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