A002547 -id:A002547 - cp's OEIS frontend

A002548 Denominators of coefficients for numerical differentiation.

1, 1, 12, 6, 180, 10, 560, 1260, 12600, 1260, 166320, 13860, 2522520, 2702700, 2882880, 360360, 110270160, 2042040, 775975200, 162954792, 56904848, 2586584, 1427794368, 892371480, 116008292400, 120470149800, 1124388064800

Offset: 2

N. J. A. Sloane

Denominator of 1 - 2*HarmonicNumber(n-1)/n. - Eric W. Weisstein, Apr 15 2004
Denominator of u(n) = sum( k=1, n-1, 1/(k(n-k)) ) (u(n) is asymptotic to 2*log(n)/n). - Benoit Cloitre, Apr 12 2003; corrected by Istvan Mezo, Oct 29 2012
Expected area of the convex hull of n points picked at random inside a triangle with unit area. - Eric W. Weisstein, Apr 15 2004

			0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, 5471/12600, ...

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).

Vincenzo Librandi, Table of n, a(n) for n = 2..250
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]
A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Triangle Point Picking
Eric Weisstein's World of Mathematics, Simplex Simplex Picking
Index entries for sequences related to the Josephus Problem

Cf. A002547, A093762.

seq(denom(Stirling1(j+2,2)/(j+2)!*2!*(-1)^j), j=0..50);

Table[Denominator[1 - 2*HarmonicNumber[n - 1]/n], {n, 2, 30}] (* Wesley Ivan Hurt, Mar 24 2014 *)

G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)).

A002547(n)/a(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)!.

More terms, GF, formula, Maple code from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 16 2007

A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

Original entry on oeis.org

0, 0, 8, 54, 1240, 70000, 7941968, 1589632128, 512918521344, 249820864339968, 174720109813751808, 168721560082538496000, 217977447876560510976000, 367117517435096337481728000, 788739873984137255456342016000, 2122296978948474538763602624512000

Offset: 0

Vaclav Kotesovec, May 15 2018

nonn

In general, for m > 1, Sum_{k=1..n-1} 1/(k*(n-k))^m is asymptotic to 2*Zeta(m)/n^m.

Seiichi Manyama, Table of n, a(n) for n = 0..150
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A052517, A302827, A002547, A002548, A304581, A304582, A304655.

List([0..20],n->Factorial(n)^3*Sum([1..n-1],k->1/(k*(n-k))^3)); # Muniru A Asiru, May 16 2018

seq(factorial(n)^3*add(1/(k*(n-k))^3,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

Table[n!^3*Sum[1/(k*(n-k))^3, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[3, x]^2, {x, 0, 20}], x] * Range[0,20]!^3

Recurrence: n^2*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)^2*(24*n^7 - 306*n^6 + 1620*n^5 - 4599*n^4 + 7516*n^3 - 7015*n^2 + 3444*n - 696)*a(n-1) - 6*(n-2)^5*(12*n^7 - 162*n^6 + 906*n^5 - 2700*n^4 + 4583*n^3 - 4378*n^2 + 2163*n - 436)*a(n-2) + 2*(n-3)^5*(n-2)^3*(24*n^7 - 342*n^6 + 2004*n^5 - 6201*n^4 + 10816*n^3 - 10497*n^2 + 5208*n - 1048)*a(n-3) - (n-4)^6*(n-3)^5*(n-2)^3*(12*n^4 - 60*n^3 + 102*n^2 - 69*n + 17)*a(n-4).

a(n) / (n!)^3 ~ 2*Zeta(3)/n^3.

A304581 Numerator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

Original entry on oeis.org

0, 0, 1, 1, 41, 13, 8009, 161, 190513, 167101, 13371157, 21857, 316786853, 371449, 52598187029, 260957190289, 129548894873, 3562512061, 295728132584141, 814542451061, 105590441859671453, 21013691164284241, 2988054680665783, 5623939943287, 1567371864703176307

Offset: 0

Vaclav Kotesovec, May 15 2018

Sum_{k=1..n-1} 1/(k*(n-k))^2 is asymptotic to Pi^2/(3*n^2) + 4*log(n)/n^3.

			0, 0, 1, 1/2, 41/144, 13/72, 8009/64800, 161/1800, 190513/2822400, ...

Eric Weisstein's World of Mathematics, Dilogarithm.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A002547, A304582, A302827.

CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 25}], x]//Numerator
Table[Sum[1/(k*(n - k))^2, {k, 1, n - 1}], {n, 0, 25}]//Numerator

A120299 Largest prime factor of Stirling numbers of first kind s(n,2) = A000254(n).

Original entry on oeis.org

3, 11, 5, 137, 7, 11, 761, 7129, 61, 863, 509, 919, 1117, 41233, 8431, 1138979, 39541, 7440427, 11167027, 18858053, 227, 583859, 467183, 312408463, 34395742267, 215087, 375035183, 4990290163, 17783, 2667653736673, 535919, 199539368321, 15088528003, 137121586897, 9059

Offset: 2

Alexander Adamchuk, Jul 11 2006

nonn

M. F. Hasler, Table of n, a(n) for n = 2..168

Cf. A000254, A002547, A001008, A002805, A006530.

Table[Max[FactorInteger[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}]]],{n,2,40}]
FactorInteger[#][[-1,1]]&/@StirlingS1[Range[3,40],2] (* Harvey P. Dale, May 10 2018 *)

A120299(n)=A006530(A000254(n)) \\ Probably A000254 can be replaced by (much smaller) A096617. - M. F. Hasler, Jul 04 2019

a(n) = Max[FactorInteger[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}]]].

a(n) = gpf(A096617(n)), where gpf = A006530 is the greatest prime factor, and A096617 is a "reduced" variant of A001008 and thus A000254. [Conjectured; true if this gpf is always > n.] - M. F. Hasler, Jul 04 2019

More terms from M. F. Hasler, Jul 04 2019

A232180 First bisection of harmonic numbers (numerators).

Original entry on oeis.org

1, 11, 137, 363, 7129, 83711, 1145993, 1195757, 42142223, 275295799, 18858053, 444316699, 34052522467, 312536252003, 9227046511387, 290774257297357, 53676090078349, 54437269998109, 2040798836801833, 2066035355155033, 85691034670497533

Offset: 1

Jean-François Alcover and Paul Curtz, Nov 20 2013

Numerator of H(2*n-1), where H(n) = Sum_{k=1..n} 1/k.

It can be noted that the second row of the Akiyama-Tanigawa transform of the fractions A232180/A232181 has a simple expression: -5/6, -9/10, -13/14, -17/18, -21/22, ... are of the form -(4*k+5)/(4*k+6).

Cf. A001008, A002547, A093158, A175441, A232181 (denominators).

[Numerator(HarmonicNumber(2*n-1)): n in [1..30]]; // Bruno Berselli, Nov 20 2013

a[n_] := HarmonicNumber[2*n-1] // Numerator; Table[a[n], {n, 1, 25}]

a(n) ~ exp(2n).

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A002548 Denominators of coefficients for numerical differentiation.

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A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

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A304581 Numerator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

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A120299 Largest prime factor of Stirling numbers of first kind s(n,2) = A000254(n).

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A232180 First bisection of harmonic numbers (numerators).

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