A002428 -id:A002428 - cp's OEIS frontend

A004041 Scaled sums of odd reciprocals: a(n) = (2n + 1)!!(Sum_{k=0..n} 1/(2*k + 1)).

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

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

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

			(arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...

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).

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)

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 *)

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

A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2n-1) )LCM(1, 3, 5, ..., 2*n-1).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

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).]
2*(1 + 1/3 + ... + 1/(2*n-1))/Pi = 2*a(n)/(A025547(n)*Pi) is the equivalent resistance between the points (0,0) and (n,n) on a 2-dimension infinite square grid of unit resistors. - Jianing Song, Apr 28 2025

Georg Fischer, Table of n, a(n) for n = 1..200 (first 39 terms from Jean-François Alcover)
MathPages, The Algebra of an Infinite Grid of Resistors
Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance?
Eric Weisstein's World of Mathematics, Jeep Problem

Cf. A025547, A075135, A002428, A350669.

a025550 n = numerator $ sum  $ map (1 %) $ take n [1, 3 ..]
-- Reinhard Zumkeller, Jan 22 2012

[&+[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

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

Table[(Total[1/Range[1,2n-1,2]])LCM@@Range[1,2n-1,2],{n,30}] (* Harvey P. Dale, Sep 09 2020 *)

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

1 + 1/3 + ... + 1/(2*n-1) = a(n)/A025547(n) = A350669(n-1)/A350670(n-1). - Jianing Song, Apr 28 2025

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

A035048 Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Robert Israel, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Transforms

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

S:= series(log(1-x)/(x^2-1), x, 101):
seq(numer(coeff(S,x,j)), j=1..100); # Robert Israel, Jun 02 2015

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 *)

a(n)=numerator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

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

A002549 Numerators of coefficients of log(1+x)/sqrt(1+x).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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).

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]

Cf. A002550, A002428.

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

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

A071968 Denominators of coefficients of expansion of arctan(x)^2 = x^2-2/3x^4+23/45x^6-44/105x^8+563/1575x^10-3254/10395*x^12+ ...

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 17 2002

Cf. A002428.

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 *)

A231121 Denominators of coefficients of expansion of arctan(x)^3.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 04 2013

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

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 *)

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.

cp's OEIS Frontend

A004041 Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).

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A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2*n-1) )*LCM(1, 3, 5, ..., 2*n-1).

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A035048 Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

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A002549 Numerators of coefficients of log(1+x)/sqrt(1+x).

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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+ ...

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A231121 Denominators of coefficients of expansion of arctan(x)^3.

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A004041 Scaled sums of odd reciprocals: a(n) = (2n + 1)!!(Sum_{k=0..n} 1/(2*k + 1)).

A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2n-1) )LCM(1, 3, 5, ..., 2*n-1).

A071968 Denominators of coefficients of expansion of arctan(x)^2 = x^2-2/3x^4+23/45x^6-44/105x^8+563/1575x^10-3254/10395*x^12+ ...