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.

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A002428 Numerators of coefficients of expansion of arctan(x)^2 = x^2 - 2/3*x^4 + 23/45*x^6 - 44/105*x^8 + 563/1575*x^10 - 3254/10395*x^12 + ...

Original entry on oeis.org

0, 1, -2, 23, -44, 563, -3254, 88069, -11384, 1593269, -15518938, 31730711, -186088972, 3788707301, -5776016314, 340028535787, -667903294192, 10823198495797, -5476065119726, 409741429887649, -103505656241356, 17141894231615609
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

|a(n)| = numerator of Sum_{k=1..n} 1/(n*(2*k-1)).
Let f(x) = (1/2)*log((1+sqrt(x))/(1-sqrt(x))) and c(n) = Integral_{x=0..1} f(x)*x^(n-1) dx, then for n>=1, c(n) = |a(n+1)|/A071968(n) and (f(x))^2 = Sum_{n>=1} c(n)*x^n. - Groux Roland, Dec 14 2010

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 89.
  • H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 313.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A071968.
Cf. A002549, A004041, A025550, A035048.

Programs

  • GAP
    List([1..30], n-> NumeratorRat( (-1)^n*Sum([1..n-1], k-> 1/((n-1)*(2*k-1))) )) # G. C. Greubel, Jul 03 2019
  • Magma
    [0] cat [Numerator((-1)^n*(&+[1/((n-1)*(2*k-1)): k in [1..n-1]])): n in [2..30]]; // G. C. Greubel, Jul 03 2019
  • Mathematica
    a[n_]:= (-1)^n*Sum[1/((n-1)*(2*k-1)), {k,1,n-1}]//Numerator; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013 *)
a[n_]:= SeriesCoefficient[ArcTan[x]^2, {x, 0, 2*n-2}]//Numerator; Table[a[n], {n, 1, 30}] (* G. C. Greubel, Jul 03 2019 *)
  • PARI
    vector(30, n, numerator((-1)^n*sum(k=1,n-1,1/((n-1)*(2*k-1))))) /* corrected by G. C. Greubel, Jul 03 2019 */
  • Sage
    [numerator((-1)^n*sum(1/((n-1)*(2*k-1)) for k in (1..n-1))) for n in (1..30)] # G. C. Greubel, Jul 03 2019

Formula

a(n) = numerator of (-1)^n * Sum_{k=1..n-1} 1/((n-1)*(2*k-1)), for n>=1. - G. C. Greubel, Jul 03 2019

Extensions

More terms from Jason Earls, Apr 09 2002
Additional comments from Benoit Cloitre, Apr 06 2002

A035047 Denominators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

Original entry on oeis.org

1, 2, 3, 4, 15, 12, 105, 24, 315, 120, 3465, 40, 45045, 280, 45045, 560, 765765, 5040, 14549535, 5040, 14549535, 55440, 334639305, 55440, 1673196525, 720720, 5019589575, 720720, 145568097675, 720720
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035048.

Programs

  • Maple
    S:= series(log(1-x)/(x^2-1), x, 101):
seq(denom(coeff(S,x,j)),j=1..100); # Robert Israel, Jun 02 2015
  • PARI
    a(n)=denominator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

Formula

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003
a(n) = denominator((-1)^(n+1)*1/2*(log(2)+(-1)^(n+1)*(gamma+1/2*(psi(1+n/2)-psi(3/2+n/2))+psi(2+n)))), with gamma the Euler-Mascheroni constant. - [Gerry Martens, Apr 28 2011]
Showing 1-2 of 2 results.

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

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