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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 + ...
(Formerly M2131 N0844)
7
0, 1, -2, 23, -44, 563, -3254, 88069, -11384, 1593269, -15518938, 31730711, -186088972, 3788707301, -5776016314, 340028535787, -667903294192, 10823198495797, -5476065119726, 409741429887649, -103505656241356, 17141894231615609 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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. - _Roland Groux_, 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

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

MATHEMATICA

a[n_]:= (-1)^n*Sum[1/((n-1)*(2*k-1)), {k, 1, n-1}]//Numerator; Table[a[n], {n, 1, 30}] (* Jean-Fran├žois 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 *)

PROG

(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 */

(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

(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

(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

CROSSREFS

Cf. A071968.

Cf. A002549, A004041, A025550, A035048.

Sequence in context: A107374 A068835 A156557 * A325145 A105440 A139831

Adjacent sequences:  A002425 A002426 A002427 * A002429 A002430 A002431

KEYWORD

sign,easy,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Jason Earls, Apr 09 2002

Additional comments from Benoit Cloitre, Apr 06 2002

STATUS

approved

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Last modified July 14 02:59 EDT 2020. Contains 335716 sequences. (Running on oeis4.)