A002545 -id:A002545

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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

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

A002547

Numerator of the n-th harmonic number H(n) divided by (n+1); a(n) = A001008(n) / ((n+1)*A002805(n)).
(Formerly M4765 N2036)

+10
6

1, 1, 11, 5, 137, 7, 363, 761, 7129, 671, 83711, 6617, 1145993, 1171733, 1195757, 143327, 42142223, 751279, 275295799, 55835135, 18858053, 830139, 444316699, 269564591, 34052522467, 34395742267, 312536252003, 10876020307, 9227046511387, 300151059037

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

OFFSET

1,3

COMMENTS

Numerators of coefficients for numerical differentiation.

REFERENCES

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).

A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.

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

M. F. Hasler, Table of n, a(n) for n = 1..2000 (first 700 terms from Alois P. Heinz)

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. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Harmonic Number

FORMULA

G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

A002547(n)/A002548(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)! - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

Numerator 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

a(n) = numerator of 2*Integral_{0..1} x^(n+1)*log(x/(1-x)) dx. - Groux Roland, May 18 2011

a(n) = numerator of A001008(n)/(n+1), since A001008(n)/A002805(n) are already in lowest terms. - M. F. Hasler, Jul 03 2019

EXAMPLE

H(n) = Sum_{k=1..n} 1/k, begins 1, 3/2, 11/6, 25/12, ... so H(n)/(n+1) begins 1/2, 1/2, 11/24, 5/12, ....

a(4) = numerator(H(4)/(4+1)) = 5.

MAPLE

H := proc(a, b) option remember; local m, p, q, r, s;

if b - a <= 1 then return 1, a fi; m := iquo(a + b, 2);

p, q := H(a, m); r, s := H(m, b); p*s + q*r, q*s; end:

A002547 := proc(n) H(1, n+1); numer(%[1]/(%[2]*(n+1))) end:

seq(A002547(n), n=1..30); # Peter Luschny, Jul 11 2019

MATHEMATICA

a[n_]:= Numerator[HarmonicNumber[n]/(n+1)])]; Table[a[n], {n, 35}] (* modified by G. C. Greubel, Jul 03 2019 *)

PROG

(PARI) h(n) = sum(k=1, n, 1/k);

vector(35, n, numerator(h(n)/(n+1))) \\ G. C. Greubel, Jul 03 2019

(PARI) A002547(n)=numerator(A001008(n)/(n+1)) \\ M. F. Hasler, Jul 03 2019

(Magma) [Numerator(HarmonicNumber(n)/(n+1)): n in [1..35]]; // G. C. Greubel, Jul 03 2019

(Sage) [numerator(harmonic_number(n)/(n+1)) for n in (1..35)] # G. C. Greubel, Jul 03 2019

(GAP) List([1..35], n-> NumeratorRat(Sum([1..n], k-> 1/k)/(n+1))) # G. C. Greubel, Jul 03 2019

CROSSREFS

Cf. A002548, A001008, A002805.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

Simpler definition from Alexander Adamchuk, Oct 31 2004

Offset corrected by Gary Detlefs, Sep 08 2011

Definition corrected by M. F. Hasler, Jul 03 2019

STATUS

approved

A002546

Denominator of Sum_{i+j+k=n; i,j,k > 0} 1/(i*j*k).
(Formerly M1110 N0424)

+10
2

1, 2, 4, 8, 15, 240, 15120, 672, 8400, 100800, 69300, 4950, 17199000, 22422400, 33633600, 201801600, 467812800, 102918816000, 410646075840, 3555377280, 215100325440, 5162407810560, 30920671782000, 190281057120, 1085315579548200, 562756226432400, 22969641895200

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

OFFSET

1,2

COMMENTS

Denominators of coefficients for numerical differentiation.

REFERENCES

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).

A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.

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..27.

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. [Annotated scanned copy]

FORMULA

G.f.: (-log(1-x))^3 (for fractions A002545(n)/A002546(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

A002545(n)/A002546(n) = 6*Stirling_1(n+3, 3)(-1)^n/(n+3)!. - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

MAPLE

seq(denom(-Stirling1(j, 3)/j!*3!*(-1)^j), j=3..50); # Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

MATHEMATICA

Denominator[Table[Sum[1/i/j/(n-i-j), {i, n-2}, {j, n-i-1}], {n, 3, 100}]] (* Ryan Propper *)

CROSSREFS

Cf. A002545.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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