A002548 -id:A002548

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A093763

Duplicate of A002548.

+20
0

1, 1, 12, 6, 180, 10, 560, 1260, 12600, 1260, 166320, 13860, 2522520, 2702700

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..14.

KEYWORD

dead

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,changed

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

A061172

Third column of Lucas bisection triangle (odd part).

+10
5

9, 120, 753, 3612, 15040, 57366, 206115, 709152, 2360943, 7659870, 24340184, 76031100, 234116493, 712166952, 2143779645, 6394719216, 18923041360, 55601888562, 162350117703, 471371537040, 1361642740059

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

OFFSET

0,1

COMMENTS

Numerator of g.f. is row polynomial Sum_{m=0..4} A061187(2,m)*x^m.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-30,45,-30,9,-1).

FORMULA

a(n) = A060924(n+2, 2).

G.f.: (3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3.

MATHEMATICA

CoefficientList[Series[(3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3, {x, 0, 50}], x] (* or *) LinearRecurrence[{9, -30, 45, -30, 9, -1}, {9, 120, 753, 3612, 15040, 57366}, 30] (* G. C. Greubel, Dec 21 2017 *)

PROG

(PARI) x='x+O('x^30); Vec((3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3) \\ G. C. Greubel, Dec 21 2017

(Magma) I:=[9, 120, 753, 3612, 15040, 57366]; [n le 6 select I[n] else 9*Self(n-1)-30*Self(n-2)+45*Self(n-3)-30*Self(n-4)+9*Self(n-5) - Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CROSSREFS

Cf. A002548, 2*A061171.

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Apr 20 2001

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

A304582

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

+10
3

1, 1, 1, 2, 144, 72, 64800, 1800, 2822400, 3175200, 317520000, 635040, 11064936960, 15367968, 2545242860160, 14609174580000, 8310997094400, 259718659200, 24319016372851200, 75058692508800, 10838475198270720000, 2389883781218693760, 374701571140216320

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

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..22.

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]//Denominator

Table[Sum[1/(k*(n - k))^2, {k, 1, n - 1}], {n, 0, 25}]//Denominator

CROSSREFS

Cf. A002548, A304581, A302827.

KEYWORD

nonn,frac,changed

AUTHOR

Vaclav Kotesovec, May 15 2018

STATUS

approved

A061174

Fifth column of Lucas bisection triangle (odd part).

+10
2

15, 545, 7043, 57560, 365045, 1970905, 9520315, 42385132, 177293730, 705980760, 2701362950, 10001654350, 36020160943, 126701700755, 436709397085, 1478813477920, 4930328078835, 16212542696607

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

OFFSET

0,1

COMMENTS

Numerator of g.f. is row polynomial sum(A061187(4,m)*x^m,m=0..7).

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n)=A060924(n+4, 4).

G.f.: (3-2*x)*(16*x^6-56*x^5+181*x^4-306*x^3+171*x^2+110*x+5)/(1-3*x+x^2)^5.

CROSSREFS

Cf. A002548, 2*A061171, A061172, 4*A061173.

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Apr 20 2001

STATUS

approved

A093762

Numerators of 1-2*HarmonicNumber(n)/(n+1).

+10
2

0, 0, 1, 1, 43, 3, 197, 499, 5471, 589, 82609, 7243, 1376527, 1530967, 1687123, 217033, 68127937, 1290761, 500679401, 107119657, 38046795, 1756445, 983477669, 622806889, 81955769933, 86074407533, 811851812797, 29280696293

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

OFFSET

1,5

COMMENTS

Expected area of the convex hull of n points picked at random inside a triangle with unit area.

LINKS

Robert Israel, Table of n, a(n) for n = 1..2296

Eric Weisstein's World of Mathematics, Triangle Point Picking

Eric Weisstein's World of Mathematics, Simplex Simplex Picking

EXAMPLE

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

MAPLE

h:= 1:

A[1]:= 0:

for n from 2 to 50 do

h:= h+1/n;

A[n]:= numer(1-2*h/(n+1));

od:

seq(A[i], i=1..50); # Robert Israel, Oct 17 2018

MATHEMATICA

Table[Numerator[1-2HarmonicNumber[n]/(n+1)], {n, 30}] (* Harvey P. Dale, Oct 10 2013 *)

PROG

(PARI) a(n) = numerator(1-2*sum(i=1, n, 1/i)/(n+1)) \\ Felix Fröhlich, Oct 17 2018

CROSSREFS

Cf. A002548.

KEYWORD

nonn,frac,changed

AUTHOR

Eric W. Weisstein, Apr 15 2004

STATUS

approved

A061175

One half of sixth column of Lucas bisection triangle (odd part).

+10
1

9, 471, 8268, 85962, 662773, 4215123, 23440212, 118073914, 551281476, 2423731704, 10148667670, 40812739230, 158644493079, 599051383561, 2206150654944, 7949311477362, 28098758599203, 97645872621753

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

OFFSET

0,1

COMMENTS

Numerator of G.f. is one half of row polynomial sum(A061187(5,m)*x^m,m=0..8).

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n)= A060924(n+5, 5)/2.

G.f.: (1+x)*(3-2*x)*(12*x^3-35*x^2+29*x+1)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^6.

CROSSREFS

Cf. A002548, 2*A061171, A061172, 4*A061173, A061174.