A002547 -id:A002547

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A002548

Denominators of coefficients for numerical differentiation.
(Formerly M4822 N2063)

+10
9

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

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

OFFSET

2,3

COMMENTS

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

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

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

FORMULA

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)!.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A002547, A093762.

KEYWORD

nonn,frac,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A304589

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

+10
5

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

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

OFFSET

0,3

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..150

Eric Weisstein's World of Mathematics, Polylogarithm.

FORMULA

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.

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,changed

AUTHOR

Vaclav Kotesovec, May 15 2018

STATUS

approved

A304581

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

+10
3

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

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

OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..24.

Eric Weisstein's World of Mathematics, Dilogarithm.

Eric Weisstein's World of Mathematics, Polylogarithm.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A002547, A304582, A302827.

KEYWORD

nonn,frac,changed

AUTHOR

Vaclav Kotesovec, May 15 2018

STATUS

approved

A120299

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

+10
2

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

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

OFFSET

2,1

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Jul 11 2006

EXTENSIONS

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

STATUS

approved

A232180

First bisection of harmonic numbers (numerators).

+10
2

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

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

OFFSET

1,2

COMMENTS

Numerator of H(2n+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).

LINKS

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

FORMULA

a(n) ~ exp(2n).

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,frac,easy

AUTHOR

Jean-François Alcover and Paul Curtz, Nov 20 2013

STATUS

approved

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