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Denominators of coefficients of expansion of arctan(x)^3.
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1, 1, 15, 945, 175, 17325, 23648625, 1576575, 7309575, 1283268987, 3360942585, 1932541986375, 135664447443525, 218461268025, 242856109621125, 27604644460267875, 4479480941961650625, 1151866527932995875, 31580724596338947904875, 809762169136896100125, 4742892704944677157875
FORMULA
Let u(n) = (-1)^n/(2*n+1) and P(n,x) = u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x), with P(0,x) = u(0). Then, the terms are the denominators of the coefficients of x^2 in each polynomial.
MATHEMATICA
a[n_] := SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}] // Denominator
(* or *) a[n_] := 3*Sum[2^(i-2)*Binomial[2*(n+1), i-1]*StirlingS1[i, 3]/i!, {i, 3, 2n+3}] // Denominator; Table[a[n], {n, 0, 20}] (* from the formula given by Ruperto Corso in A002429 *) Take[Denominator[CoefficientList[Series[ArcTan[x]^3, {x, 0, 50}], x] ], {4, -1, 2}] (* Harvey P. Dale, Apr 07 2017 *)
a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2*n-1) )*LCM(1, 3, 5, ..., 2*n-1).
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1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427
COMMENTS
Or, numerator of 1/1 + 1/3 + ... + 1/(2n-1) up to a(38).
Following similar remark by T. D. Noe in A025547, this coincides with f(n) = numerator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 18048708369314455836683437302413, f(39) = 1640791669937677803334857936583. Note that f(n) = numerator(digamma(n+1/2)/2 + log(2) + euler_gamma/2). - Paul Barry, Aug 19 2005 [See A350669(n-1).]
MAPLE
a:= n-> (f-> add(1/p, p=f)*ilcm(f[]))([2*i-1$i=1..n]):
MATHEMATICA
Table[(Total[1/Range[1, 2n-1, 2]])LCM@@Range[1, 2n-1, 2], {n, 30}] (* Harvey P. Dale, Sep 09 2020 *)
PROG
(Haskell)
a025550 n = numerator $ sum $ map (1 %) $ take n [1, 3 ..]
(Magma) [&+[1/d: d in i]*Lcm(i) where i is [1..2*n-1 by 2]: n in [1..21]]; // Bruno Berselli, Apr 16 2015
Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
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1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500
COMMENTS
n-th elementary symmetric function of the first n+1 odd positive integers.
Also the determinant of the n X n matrix given by m(i,j) = 2*i + 2 = if i = j and otherwise 1. For example, Det[{{4, 1, 1, 1, 1, 1}, {1, 6, 1, 1, 1, 1}, {1, 1, 8, 1, 1, 1}, {1, 1, 1, 10, 1, 1}, {1, 1, 1, 1, 12, 1}, {1, 1, 1, 1, 1, 14}}] = 264207 = a(6). - John M. Campbell, May 20 2011
FORMULA
a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
a(n) is coefficient of x^(2*n+2) in (arctanh x)^2, multiplied by (n + 1)*(2*n + 1)!!.
a(n) = Sum_{i=k+1..n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 1, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001
a(n) ~ 2^(1/2)*log(n)*n*(2n/e)^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/2*(1 - 2*x)^(-3/2)*(2 - log(1 - 2*x)). - Vladeta Jovovic, Feb 19 2003 For n >= 1, a(n-1) = 2^(n-1)*n!*(Sum_{k=0..n-1} (-1)^k*binomial(1/2, k)/(n - k)). - Milan Janjic, Dec 14 2008
EXAMPLE
(arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
MATHEMATICA
Table[(-1)^(n + 1)* Sum[(-2)^(n - k) k (-1)^(n - k) StirlingS1[n + 1, k + 1], {k, 0, n}], {n, 1, 18}] (* Zerinvary Lajos, Jul 08 2009 *) FunctionExpand@Table[(2 n + 1)!! (Log[4] + HarmonicNumber[n + 1/2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 13 2016 *)
CROSSREFS
Equals second left hand column of A028338 triangle. Equals second right hand column of A109692 triangle. Equals second left hand column of A161198 triangle divided by 2. (End)
AUTHOR
Joe Keane (jgk(AT)jgk.org)
Numerators of coefficients of log(1+x)/sqrt(1+x).
(Formerly M5128 N2223)
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1, 1, 23, 11, 563, 1627, 88069, 1423, 1593269, 7759469, 31730711, 46522243, 3788707301, 2888008157, 340028535787, 41743955887, 10823198495797, 2738032559863, 409741429887649, 25876414060339, 17141894231615609
REFERENCES
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).
PROG
(PARI) x='x+O('x^66); abs(apply(t->numerator(t), Vec(log(1+x)/sqrt(1+x)))) \\ Joerg Arndt, Apr 18 2014
Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/ A002805.
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1, 1, 4, 3, 23, 11, 176, 25, 563, 137, 6508, 49, 88069, 363, 91072, 761, 1593269, 7129, 31037876, 7381, 31730711, 83711, 744355888, 86021, 3788707301, 1145993, 11552032628, 1171733, 340028535787, 1195757
FORMULA
a(n) = Numerator[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i,{i,1,k}],{k,1,n}]]. - Alexander Adamchuk, Jul 07 2006 a(n) = numerator((-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
MAPLE
S:= series(log(1-x)/(x^2-1), x, 101):
seq(numer(coeff(S, x, j)), j=1..100); # Robert Israel, Jun 02 2015
MATHEMATICA
Numerator[Table[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i, {i, 1, k}], {k, 1, n}], {n, 1, 50}]] (* Alexander Adamchuk, Jul 07 2006 *)
PROG
(PARI) a(n)=numerator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)), n))
Denominators 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+ ...
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1, 1, 3, 45, 105, 1575, 10395, 315315, 45045, 6891885, 72747675, 160044885, 1003917915, 21751554825, 35137127025, 2183521465125, 4512611027925, 76714387474725, 40613499251325, 3172365552631275
MATHEMATICA
a[n_] := (-1)^(n+1)*Sum[1/(n*(2*k-1)), {k, 1, n}] // Denominator; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 04 2013 *)
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