A002428 -id:A002428

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A004041

Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).

+10
18

1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..403

J. Courtiel, K. Yeats, Terminal chords in connected chord diagrams, arXiv:1603.08596 [math.CO], 2016; e.g.f. in Remark 1 B_1(z).

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

Sum_{n>=1} a(n-1)/(n!*n*2^n) = (Pi/2)^2. - Philippe Deléham, Aug 12 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

Recurrence: a(n) = 4*n*a(n-1) - (2*n - 1)^2*a(n-2). - Vladimir Reshetnikov, Oct 13 2016

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

Cf. A000254, A024199, A049034.

Cf. A002428.

From Johannes W. Meijer, Jun 08 2009: (Start)

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)

KEYWORD

nonn

AUTHOR

Joe Keane (jgk(AT)jgk.org)

STATUS

approved

A025550

a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2*n-1) )*LCM(1, 3, 5, ..., 2*n-1).

+10
12

1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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).]

LINKS

Georg Fischer, Table of n, a(n) for n = 1..200 (first 39 terms from Jean-François Alcover)

Eric Weisstein's World of Mathematics, Jeep Problem

MAPLE

a:= n-> (f-> add(1/p, p=f)*ilcm(f[]))([2*i-1$i=1..n]):

seq(a(n), n=1..40); # Alois P. Heinz, Apr 16 2015

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 ..]

-- Reinhard Zumkeller, Jan 22 2012

(PARI) a(n)=my(v=vector(n, i, 2*i-1)); sum(i=1, #v, 1/v[i])*lcm(v) \\ Charles R Greathouse IV, Feb 28 2013

(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

CROSSREFS

Cf. A025547, A075135, A002428, A350669.

KEYWORD

nonn,easy,nice,frac

AUTHOR

Clark Kimberling

EXTENSIONS

Value of a(39) corrected by Jean-François Alcover, Apr 16 2015

STATUS

approved

A035048

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

p^2 divides a(2p-2) for prime p>3. a(2p-2)/p^2 = A061002(n) = A001008(p-1)/p^2 for prime p>2. - Alexander Adamchuk, Jul 07 2006

LINKS

Robert Israel, Table of n, a(n) for n = 1..2000

N. J. A. Sloane, Transforms

FORMULA

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003

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))

CROSSREFS

Cf. A035047, A002428, A001008, A058313, A061002.

KEYWORD

nonn,easy,frac

AUTHOR

N. J. A. Sloane.

STATUS

approved

A002549

Numerators of coefficients of log(1+x)/sqrt(1+x).
(Formerly M5128 N2223)

+10
2

1, 1, 23, 11, 563, 1627, 88069, 1423, 1593269, 7759469, 31730711, 46522243, 3788707301, 2888008157, 340028535787, 41743955887, 10823198495797, 2738032559863, 409741429887649, 25876414060339, 17141894231615609

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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).

LINKS

Table of n, a(n) for n=1..21.

W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).

W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]

PROG

(PARI) x='x+O('x^66); abs(apply(t->numerator(t), Vec(log(1+x)/sqrt(1+x)))) \\ Joerg Arndt, Apr 18 2014

CROSSREFS

Cf. A002550, A002428.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Benoit Cloitre, Mar 29 2002

a(15) corrected by Sean A. Irvine, Mar 24 2014

a(18) corrected by Sean A. Irvine, Apr 17 2014

STATUS

approved

A071968

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+ ...

+10
2

1, 1, 3, 45, 105, 1575, 10395, 315315, 45045, 6891885, 72747675, 160044885, 1003917915, 21751554825, 35137127025, 2183521465125, 4512611027925, 76714387474725, 40613499251325, 3172365552631275

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

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 *)

CROSSREFS

Cf. A002428.

KEYWORD

nonn,easy,frac

AUTHOR

N. J. A. Sloane, Jun 17 2002

STATUS

approved

A231121

Denominators of coefficients of expansion of arctan(x)^3.

+10
0

1, 1, 15, 945, 175, 17325, 23648625, 1576575, 7309575, 1283268987, 3360942585, 1932541986375, 135664447443525, 218461268025, 242856109621125, 27604644460267875, 4479480941961650625, 1151866527932995875, 31580724596338947904875, 809762169136896100125, 4742892704944677157875

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

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 *)

CROSSREFS

Cf. A002428, A002429, A071968, A140749, A141904, A142048, A081051.

KEYWORD

nonn,frac

AUTHOR

Jean-François Alcover and Paul Curtz, Nov 04 2013

STATUS

approved

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