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.

A052468 Numerators in the Taylor series for arccosh(x) - log(2*x).

Original entry on oeis.org

1, 3, 5, 35, 63, 77, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 646323, 300540195, 583401555, 756261275, 4418157975, 6892326441, 22427411435, 263012370465, 514589420475, 2687300306925, 15801325804719, 61989816618513, 121683714103007
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

A055786 is the preferred version of this sequence.

Examples

			i*Pi/2 - arccosh(x) = i*x + (1/6)*i*x^3 + (3/40)*i*x^5 + (5/112)*i*x^7 + (35/1152)*i*x^9 + (63/2816)*i*x^11 + (231/13312)*i*x^13 + (143/10240)*i*x^15 + (6435/557056)*i*x^17 + ...
0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A052468/A052469

Links

Crossrefs

See A055786 for further information.
a(n)/A052469(n) = (1/(2*n))*A001790(n)/A046161(n) for n=>1.
Equals A162441(n+1)/(2n+1) for n=>1. - Johannes W. Meijer, Jul 06 2009

Programs

  • GAP
    List([1..30], n-> NumeratorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019
  • Magma
    [Numerator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017
  • Mathematica
    a [n_]:=Numerator[(2 n - 1)! / (2^(2 n) n!^2)]; Array[a, 40] (* Vincenzo Librandi, Jul 10 2017 *)
  • PARI
    {a(n) = numerator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019
  • Sage
    [numerator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

Formula

a(n)/A052469(n) = A001147(n)/(A000165(n)*2*n). E.g., a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ).
a(n) = numerator((2*n-1)!/(4^n * (n!)^2)). - Johannes W. Meijer, Jul 06 2009
Let z(n) = 2*(2*n+1)!*4^(-n-1)/((n+1)!)^2, then a(n) = numerator(z(n)), A162442(n) = denominator(z(n)), and z(n) = 1/(n+1) - Sum_{k=0..n}(-1)^k*binomial(n,k)*z(k). - Groux Roland, Jan 04 2011
a(n) = numerator(binomial(2n,n)/(n*2^(2n-1))). - Daniel Suteu, Oct 30 2017

Extensions

Updated by Frank Ellermann, May 22 2011
Cross-references edited by Johannes W. Meijer, Jul 05 2009

A052469 Denominators in the Taylor series for arccosh(x) - log(2*x).

Original entry on oeis.org

4, 32, 96, 1024, 2560, 4096, 28672, 524288, 1179648, 5242880, 11534336, 100663296, 218103808, 939524096, 134217728, 68719476736, 146028888064, 206158430208, 1305670057984, 2199023255552, 7696581394432, 96757023244288, 202310139510784, 1125899906842624
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Examples

			arccosh(x) = log(2x) - 1/(4*x^2) - 3/(32*x^4) - 5/(96*x^6) - ... for x>1.

References

  • Bronstein-Semendjajew, sprawotchnik po matematikje, 6th Russian ed. 1956, ch. 4.2.6.

Links

Crossrefs

Cf. A002595.

Programs

  • GAP
    List([1..30], n-> DenominatorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019
  • Magma
    [Denominator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017
  • Mathematica
    a[n_] := Denominator[(2*n-1)!/(2^(2*n)*n!^2)]; Array[a, 21] (* Jean-François Alcover, May 17 2017 *)
  • PARI
    {a(n) = denominator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019
  • Sage
    [denominator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

Formula

A052468(n) / a(n) = A001147(n) / ( A000165(n) *2*n )
From Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = denom((2*n-1)!/( 4^n * (n!)^2)).
Equals 2*A162442(n+1) for n >= 1.
A052468(n)/a(n) = (1/(2*n))*A001790(n)/A046161(n) for n>=1.
(End)

Extensions

Updated by Frank Ellermann, May 22 2001

A162441 Numerators of the column sums of the EG1 matrix coefficients.

Original entry on oeis.org

3, 15, 35, 315, 693, 1001, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 20036013, 9917826435, 20419054425, 27981667175, 172308161025, 282585384081, 964378691705, 11835556670925, 24185702762325
Offset: 2

Views

Author

Johannes W. Meijer, Jul 06 2009

Keywords

Comments

For the definition of the EG1 matrix coefficients see A162440.
We define the columns sums by cs(n) = sum(EG1[2*m-1,n], m = 1.. infinity) for n => 2.
The row sums of the EG1 matrix follow the same pattern as those of its even counterpart the EG2 matrix, see A161739 and the formulas.

Crossrefs

Equals (2*n-1)*A052468(n-1)
Cf. A162440 and A162442 [denom(cs(n))].
Cf. A161739 (RSEG2 triangle), A001803 and A046161.

Formula

a(n) = numer(cs(n)) and denom(cs(n)) = A162442(n) with cs(n) = (2^(2-2*n)/(n-1))*((2*n-1)!/((n-1)!^2)).
cs(n) = 2*EG1[ -1,n]/(n-1) with EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2).
cs(n) = (1/(n-1))*A001803(n-1)/A046161(n-1) for n=>2.
rs(2*m-1,p=0) = sum((n^p)*EG1(2*m-1,n), n = 1..infinity) = 2*zeta(2*m-2) for m =>2.
Showing 1-3 of 3 results.

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