A002548 -id:A002548 - cp's OEIS frontend

A093763 Duplicate of A002548.

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

Offset: 1

dead

A002547 Numerator of the n-th harmonic number H(n) divided by (n+1); a(n) = A001008(n) / ((n+1)*A002805(n)).

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

Offset: 1

N. J. A. Sloane

Numerators of coefficients for numerical differentiation.

			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.

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).

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

Cf. A002548, A001008, A002805.

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

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

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

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

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

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

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

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

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

A061172 Third column of Lucas bisection triangle (odd part).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Apr 20 2001

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

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).

Cf. A002548, 2*A061171.

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

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 *)

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

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.

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.

A304582 Denominator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 15 2018

			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. A002548, A304581, A302827.

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

A061174 Fifth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Apr 20 2001

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

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

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.

A093762 Numerators of 1-2*HarmonicNumber(n)/(n+1).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 15 2004

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

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

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

Cf. A002548.

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

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

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

A061175 One half of sixth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Apr 20 2001

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

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

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.

cp's OEIS Frontend

A093763 Duplicate of A002548.

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A002547 Numerator of the n-th harmonic number H(n) divided by (n+1); a(n) = A001008(n) / ((n+1)*A002805(n)).

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A061172 Third column of Lucas bisection triangle (odd part).

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

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A304582 Denominator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

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A061174 Fifth column of Lucas bisection triangle (odd part).

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A093762 Numerators of 1-2*HarmonicNumber(n)/(n+1).

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A061175 One half of sixth column of Lucas bisection triangle (odd part).

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