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.

A090816 a(n) = (3*n+1)!/((2*n)! * n!).

Original entry on oeis.org

1, 12, 105, 840, 6435, 48048, 352716, 2558160, 18386775, 131231100, 931395465, 6580248480, 46312074900, 324897017760, 2272989850440, 15863901576864, 110487596768703, 768095592509700, 5330949171823275, 36945070220658600, 255702514854135195, 1767643865751234240
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)excite.com), Feb 11 2004

Keywords

Examples

			a(1) = 4!/(2!*1!) = 24/2 = 12.

Links

Crossrefs

Cf. A002457, A005809, A016777, A045721, A090957, A090969.
Halfdiagonal of triangle A003506.

Programs

  • Magma
    [Factorial(3*n+1)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019
  • Maple
    a:=n-> binomial(3*n+1,2*n)*(n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jul 31 2006
  • Mathematica
    f[n_] := 1/Integrate[(x^2 - x^3)^n, {x, 0, 1}]; Table[ f[n], {n, 0, 19}] (* Robert G. Wilson v, Feb 18 2004 *)
Table[1/Beta[2*n+1,n+1], {n,0,20}] (* G. C. Greubel, Feb 03 2019 *)
  • PARI
    a(n)=if(n<0,0,(3*n+1)!/(2*n)!/n!) /* Michael Somos, Feb 14 2004 */
  • PARI
    a(n)=if(n<0,0,1/subst(intformal((x^2-x^3)^n),x,1)) /* Michael Somos, Feb 14 2004 */
  • Sage
    [1/beta(2*n+1,n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019

Formula

a(n) = A005809(n) * A016777(n).
a(n) = 1/(Integral_{x=0..1} (x^2 - x^3)^n dx).
G.f.: (((8 + 27*z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) + 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)) - 3*i*sqrt(3)*sqrt(4 - 27*z)*sqrt(z)*(1/(4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(1/3) - 1/(4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(1/3)))*8^(1/3))/(2*(4 - 27*z)^(3/2)), where i is the imaginary unit. - Karol A. Penson, Feb 06 2024
a(n) = Sum_{k = 0..n} (-1)^(n+k) * (2*n + 2*k + 1)*binomial(2*n+k, k). This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (2*n + 2*k + 1) * binomial(2*n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+2)*n+1, 2*n). Cf. A002457 and A306290. - Peter Bala, Nov 02 2024
From Amiram Eldar, Dec 09 2024: (Start)
Sum_{n>=0} 1/a(n) = f(c) = 1.09422712102982285131..., where f(x) = (x*(x-1)/(3*x-1)) * ((3/2)*log(abs(x/(x-1))) + ((3*x-2)/sqrt(3*x^2-4*x)) * (arctan(x/sqrt(3*x^2-4*x)) + arctan((2-x)/sqrt(3*x^2-4*x)))), and c = 2/3 + (1/3)*((25+3*sqrt(69))/2)^(-1/3) + (1/3)*((25+3*sqrt(69))/2)^(1/3).
Sum_{n>=0} (-1)^n/a(n) = f(d) = 0.92513707957813718109..., where f(x) is defined above, and d = 2/3 - (1/3)*((29+3*sqrt(93))/2)^(-1/3) - (1/3)*((29+3*sqrt(93))/2)^(1/3).
Both formulas are from Batir (2013). (End)

Extensions

New definition from Vladeta Jovovic, Feb 12 2004

A090969 a(n) = 1/Integral_{x=0..1} (x^5 - x^6)^n.

Original entry on oeis.org

1, 42, 858, 15504, 265650, 4417686, 72068304, 1160068104, 18490100706, 292486494300, 4599035681526, 71963547329856, 1121519754006288, 17419158268943970, 269767427275060200, 4167406330765934256
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)excite.com), Feb 29 2004

Keywords

Links

Crossrefs

Cf. A002457, A004355, A016921, A090816, A090957.

Programs

  • Magma
    [Factorial(6*n+1)/(Factorial(n)*Factorial(5*n)): n in [0..20]]; // G. C. Greubel, Feb 03 2019
  • Maple
    seq(factorial(6*n+1)/(factorial(n)*factorial(5*n)), n = 0 .. 16); # Emeric Deutsch, Jun 29 2009
  • Mathematica
    Table[1/Beta[5*n+1, n+1], {n,0,20}] (* G. C. Greubel, Feb 03 2019 *)
  • PARI
    vector(20, n, n--; (6*n+1)!/(n!*(5*n)!)) \\ G. C. Greubel, Feb 03 2019
  • Sage
    [1/beta(5*n+1,n+1) for n in range(20)] # G. C. Greubel, Feb 03 2019

Formula

a(n) = A016921(n)*A004355(n). - R. J. Mathar, Jun 21 2009
a(n) = 1/B(5*n+1,n+1) = (6*n+1)!/(n! * (5*n)!), where B(p,q) is Euler's beta function (basically identical with R. J. Mathar's comment). - Emeric Deutsch, Jun 29 2009
a(n) ~ 2^(6*n+1) * 3^(6*n+3/2) * sqrt(n) / (sqrt(Pi) * 5^(5*n+1/2)). - Vaclav Kotesovec, Aug 15 2017

Extensions

Extended by Emeric Deutsch, Jun 29 2009

A306290 a(n) = 1/(Integral_{x=0..1} (x^3 - x^4)^n dx).

Original entry on oeis.org

1, 20, 252, 2860, 30940, 325584, 3364900, 34337160, 347103900, 3483301360, 34754081648, 345120260940, 3413758188932, 33655718658800, 330869721936600, 3244839440755920, 31754250910172700, 310165459118369712, 3024542552887591120, 29449493278116018800, 286360607519186119920
Offset: 0

Views

Author

G. C. Greubel, Feb 03 2019

Keywords

Links

Crossrefs

Cf. A002457, A005810, A016813, A090816, A090957, A090969.

Programs

  • GAP
    List([0..30], n -> Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)));
  • Magma
    [Factorial(4*n+1)/(Factorial(n)*Factorial(3*n)): n in [0..20]];
  • Mathematica
    Table[1/Beta[3*n+1, n+1], {n, 0, 20}]
  • PARI
    vector(20, n, n--; (4*n+1)!/(n!*(3*n)!))
  • Sage
    [1/beta(3*n+1,n+1) for n in range(20)]

Formula

a(n) = 1/Beta(3*n+1,n+1) = (4*n+1)!/(n! * (3*n)!).
a(n) = Sum_{k = 0..n} (-1)^(n+k) * (3*n + 2*k + 1)*binomial(3*n+k, k). This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (3*n + 2*k + 1) * binomial(3*n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+3)*n+1, 3*n). - Peter Bala, Nov 02 2024
From Amiram Eldar, Dec 09 2024: (Start)
a(n) = (4*n + 1) * binomial(4*n, n) = A016813(n) * A005810(n).
Formulas from Batir (2013):
Sum_{n>=0} 1/a(n) = f(c) = 1.05435362585114283076..., where f(x) = (x*(x^2-1)/(2*(2*x^2+1))) * log(abs((x+1)/(x-1))) + ((x-1)*(x^3+1)/(4*x*(2*x^2+1))) * sqrt(x/(x-2)) * (arctan(sqrt(x/(x-2))) + arctan(((3-x)/(x+1))*sqrt(x/(x-2)))) + ((x+1)*(x^3-1)/(4*x*(2*x^2+1))) * sqrt(x/(x+2)) * (arctan(((x+3)/(x-1))*sqrt(x/(x+2))) - arctan(sqrt(x/(x+2)))), and c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)).
Sum_{n>=0} (-1)^n/a(n) = f(d) = 0.953648123517883351708..., where f(x) is defined above, and d = sqrt(1 + 16*(2/(3*(9+sqrt(849))))^(1/3) - 2*(2/3)^(2/3)*(9+sqrt(849))^(1/3)). (End)
Showing 1-3 of 3 results.

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