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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

%I M2131 N0844 #45 Feb 27 2022 15:51:57

%S 0,1,-2,23,-44,563,-3254,88069,-11384,1593269,-15518938,31730711,

%T -186088972,3788707301,-5776016314,340028535787,-667903294192,

%U 10823198495797,-5476065119726,409741429887649,-103505656241356,17141894231615609

%N 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 + ...

%C |a(n)| = numerator of Sum_{k=1..n} 1/(n*(2*k-1)).

%C 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

%D 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.

%D H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 313.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A002428/b002428.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

%t 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 *)

%t a[n_]:= SeriesCoefficient[ArcTan[x]^2, {x, 0, 2*n-2}]//Numerator; Table[a[n], {n, 1, 30}] (* _G. C. Greubel_, Jul 03 2019 *)

%o (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 */

%o (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

%o (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

%o (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

%Y Cf. A071968.

%Y Cf. A002549, A004041, A025550, A035048.

%K sign,easy,frac

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Jason Earls_, Apr 09 2002

%E Additional comments from _Benoit Cloitre_, Apr 06 2002