A027447 -id:A027447 - cp's OEIS frontend

A027451 First diagonal of A027447.

1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, 6350400, 5336100, 768398400, 662547600, 577152576, 2029052025, 519437318400, 463325262400, 150117385017600, 135480939978384, 122885206329600, 111967718990400, 54192375991353600, 49770428644836900

Offset: 1

Olivier Gérard

nonn

Equals the denominators of MN(z;n)/(n!)^2 for n =>1, see A162990. - Johannes W. Meijer, Jul 21 2009

It appears that a(n) = denominator of n^2*sum(1/k^2,k=1..n). - Gary Detlefs, May 29 2010

From Johannes W. Meijer, Jul 21 2009: (Start)
Equals A002944(n)^2.
Equals A001044(n-1)/A025527(n)^2.
(End)

a[n_] := (LCM @@ Range[n]/n)^2; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 05 2013, after Johannes W. Meijer *)

Numerators of sequence a[ n, n ] in (a[ i, j ])^3 where a[ i, j ] = 1/i if j<=i, 0 if j>i.

a(n) = (lcm($1..n)/n)^2. - Johannes W. Meijer, Jul 21 2009

More terms from Sean A. Irvine, Nov 04 2019

A027460 Second column of A027447.

Original entry on oeis.org

1, 19, 115, 3799, 4669, 268921, 1227199, 12335311, 13560283, 1781891563, 1916566873, 345677548297, 366562289317, 386631866845, 1623817435135, 490825547301775, 511634386192615, 191966753701218895, 199005637912597915, 205832222367859915, 212460475248691315

Offset: 2

Olivier Gérard

nonn

Numerators of sequence a(2, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.

More terms from Sean A. Irvine, Nov 04 2019

A329108 First column of A027447.

Original entry on oeis.org

1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 788807993197, 817534859365, 3379894237615, 1007162177631295, 1036310871245335, 384227195120585215, 393975698131531915, 403382871919715515

Offset: 1

Sean A. Irvine, Nov 04 2019

nonn

This sequences was originally A027459, but differs from the current version of A027459 for n >= 14. This difference arises because the rational numbers in the matrix defining A027447 (and hence this sequence) are not reduced to the lowest common denominator, whereas the values in A027459 are.

Cf. A027447, A027459, A027460.

A027452 Second diagonal of A027447.

Original entry on oeis.org

7, 19, 37, 549, 364, 12700, 38025, 265825, 212464, 21019824, 17507700, 2502630900, 2144458800, 1858077936, 6501984489, 1657735504425, 1473342270400, 475835044484800, 428209840762704, 387395612954064, 352152440452800, 170078965146417600, 155897164961237700

Offset: 2

Olivier Gérard

nonn

Numerators of sequence a(n, n-1) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.

More terms from Sean A. Irvine, Nov 04 2019

A027453 Third diagonal of A027447.

Original entry on oeis.org

85, 115, 1489, 919, 30676, 89125, 609625, 479269, 46805584, 38580444, 5467738500, 4651663500, 4005934416, 13944602529, 3539108614041, 3132881376625, 1008229438216000, 904461877983664, 815945004944784, 739827337966704, 356490237457224000, 326078767183396500

Offset: 3

Olivier Gérard

nonn

Numerators of sequence a(n, n-2) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.

More terms from Sean A. Irvine, Nov 04 2019

A027461 Third column of A027447.

Original entry on oeis.org

4, 37, 1489, 2059, 128431, 621139, 6527971, 7435423, 1005856303, 1108515013, 204122194357, 220369102177, 236110035385, 1005497693095, 307715924645815, 324354836175055, 122932720987212535, 128616795798807715, 134153085587608915, 139549693559545915

Offset: 3

Olivier Gérard

nonn

Numerators of sequence a(3, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.

More terms from Sean A. Irvine, Nov 04 2019

A027448 Triangle read by rows: 4th power of the lower triangular mean matrix (M[i,j] = 1/i for i <= j).

Original entry on oeis.org

1, 15, 1, 575, 65, 8, 5845, 865, 175, 27, 874853, 153713, 39743, 9963, 1728, 1009743, 200403, 60333, 19153, 5368, 1000, 389919909, 84873489, 28400079, 10419739, 3681784, 1105000, 216000, 3449575767, 807843807, 292420227

Offset: 1

Olivier Gérard

			Table starts:
          1
         15           1
        575          65           8
       5845         865         175          27
     874853      153713       39743        9963        1728
    1009743      200403       60333       19153        5368        1000

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A027446 (square of M), A027447 (cube of M).

Rows:= 10:
M:= Matrix(Rows,Rows,(i,j) -> `if`(i>=j,1/i,0)):
B:= M^4:
L:= [seq(ilcm(seq(denom(B[i,j]),j=1..i)),i=1..Rows)]:
seq(seq(B[i,j]*L[i],j=1..i),i=1..Rows); # Robert Israel, Oct 05 2019

rows = 8; m = Table[ If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}]; m4 = m.m.m.m; Table[ fracs = m4[[i]]; nums = fracs // Numerator; dens = fracs // Denominator; lcm = LCM @@ dens; Table[ nums[[j]]*lcm/dens[[j]], {j, 1, i}], {i, 1, rows}] // Flatten (* Jean-François Alcover, Mar 05 2013 *)

A027448_upto(n)={my(M=matrix(n, n, i, j, (j<=i)/i)^4); vector(n,r,M[r,1..r]*denominator(M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

Let M be the lower triangular matrix with entries M[i,j] = 1/i for 1<=j<=i, and B = M^4. Then a(i,j) = B(i,j)*lcm(denom(B(i,1)),...,denom(B(i,i))). - Robert Israel, Oct 05 2019

That is, the fractions in M^4 are written using the least common denominator before taking the numerators. - M. F. Hasler, Nov 05 2019

Edited by Robert Israel, Oct 05 2019

A027446 Triangle read by rows: square of the lower triangular mean matrix.

Original entry on oeis.org

1, 3, 1, 11, 5, 2, 25, 13, 7, 3, 137, 77, 47, 27, 12, 147, 87, 57, 37, 22, 10, 1089, 669, 459, 319, 214, 130, 60, 2283, 1443, 1023, 743, 533, 365, 225, 105, 7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 847, 532, 252

Offset: 1

Olivier Gérard

Numerators of nonzero elements of A^2, written as rows using the least common denominator, where A[i,j] = 1/i if j <= i, 0 if j > i. [Edited by M. F. Hasler, Nov 05 2019]

			Triangle starts
     1
     3,    1
    11,    5,    2
    25,   13,    7,    3
   137,   77,   47,   27,   12
   147,   87,   57,   37,   22,   10
  1089,  669,  459,  319,  214,  130,  60
  2283, 1443, 1023,  743,  533,  365, 225, 105
  7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280
  ... - _Joerg Arndt_, Mar 29 2013

L. Bendersky, Sur la fonction gamma généralisée, Acta Math. 61 (1933), p. 263-322. See p. 295.

The row sums give A081528(n), n>=1.
The column sequences give A025529, A027457, A027458 for j=1..3.
The diagonal sequences give A002944, A027449, A027450.
Cf. A027447, A027448.

rows = 10;
M = MatrixPower[Table[If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}], 2];
T = Table[M[[n]]*LCM @@ Denominator[M[[n]]], {n, 1, rows}];
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 05 2013, updated May 06 2022 *)

A027446_upto(n)={my(M=matrix(n, n, i, j, (j<=i)/i)^2); vector(n,r,M[r,1..r]*denominator(M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

The rational matrix A^2, where the matrix A has elements a[i,j] = 1/A002024(i,j), is equal to A119947(i,j)/A119948(i,j).

a(i,j) = lcm(seq(A119948(i,m),m=1..i))*A119947(i,j)/A119948(i,j), 1 <= j =< i and zero otherwise.

Edited by M. F. Hasler, Nov 05 2019

A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.

Original entry on oeis.org

1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 112686856171, 3336876977, 96568406789, 28776062218037, 29608882035581, 1568274265798307, 11256448518043769

Offset: 1

Olivier Gérard

nonn

Originally defined as the first column of A027447, but now contains numerator in reduced form (cf. A329108). - Sean A. Irvine, Nov 04 2019
Numerators of the binomial transform of (-1)^n/(n+1)^3. The matrix a[i,j] below is the product of the binomial matrix and the matrix with general term binomial(i,j)(-1)^(i-j)/(i+1)^3. - Paul Barry, Aug 06 2004
From Alexander Adamchuk, Jan 02 2007 [edited by Jon E. Schoenfield, Mar 08 2015]: (Start)
Also a(n) is a numerator of S(n) = Sum_{k=1..n} H(k)/k, where H(k) is the k-th harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k).
S(n) = Sum_{k=1..n} H(k)/k = 1/2*(H(n)^2 + H(n,2)), where H(n,2) = Sum_{i=1..n} 1/i^2 = A007406(n)/A007407(n).
p divides a(p-1) and a(p-2) for prime p>3. a(n) is prime for n = {2, 7, 26, 31, 43, 53, 68, 80, 91, 123, 175, 236, 458, ...}. (End)
The n-fold repeated integral of (1/2)*log(x)^2 (all improper integrals with the lower limits of integration equal to 0) = x^n/n! * ( (1/2)*log(x)^2 - H(n)*log(x) + Sum_{k = 1..n} H(k)/k ). - Peter Bala, Feb 17 2022

			(a[ i,j ])^3 = MATRIX([[1, 0, 0, 0, 0], [7/8, 1/8, 0, 0, 0], [85/108, 19/108, 1/27, 0, 0], [415/576, 115/576, 37/576, 1/64, 0], [12019/18000, 3799/18000, 1489/18000, 61/2000, 1/125]]), n = 5.

Alexander Adamchuk, Table of n, a(n) for n = 1..30
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A007406, A007407, A027447, A329108.

[Numerator(&+[HarmonicNumber(k)/k:k in [1..n]]):n in [1..20]]; // Marius A. Burtea, Nov 05 2019

Table[Numerator[Sum[Sum[1/i,{i,1,k}]/k,{k,1,n}]],{n,1,30}] (* Alexander Adamchuk, Jan 02 2007 *)
With[{nn=20},Accumulate[HarmonicNumber[Range[nn]]/Range[nn]]]//Numerator (* Harvey P. Dale, Feb 26 2023 *)

Numerators of sequence a(1, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.
Numerators of (Wolstenholme(n, 1)^2 + Wolstenholme(n, 2))/(2*n)= ((gamma+Psi(n+1))^2 + Pi^2/6 - Psi(1, n+1))/(2*n), where Wolstenholme(n, m) = Sum_{i=1..n} 1/i^m. - Vladeta Jovovic, Aug 09 2002
a(n) = numerator(Sum_{k=1..n} ((Sum_{i=1..k} 1/i)/k)). - Alexander Adamchuk, Jan 02 2007

Corrected by Vladeta Jovovic, Aug 09 2002

A119932 Triangle of denominators of the cube of a certain lower triangular matrix.

Original entry on oeis.org

1, 8, 8, 108, 108, 27, 576, 576, 576, 64, 18000, 18000, 18000, 2000, 125, 21600, 21600, 21600, 21600, 5400, 216, 1234800, 1234800, 1234800, 1234800, 308700, 12348, 343, 5644800, 5644800, 5644800, 5644800

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of the corresponding numerators is A119935.

See a comment under A119935 on the relation to the triangle A027447.

W. Lang: First ten rows and rationals.

a(i, j)=1/A002024(i, j), i>=1, j<=i.

Row sums give A119933. Row sums of the triangle of rationals are identical 1.

a(i,j)= denominator(r(i,j)) with r(i,j):=(A^3)[i,j], where the lower triangular matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i.

cp's OEIS Frontend

A027451 First diagonal of A027447.

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A027460 Second column of A027447.

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A329108 First column of A027447.

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A027452 Second diagonal of A027447.

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A027453 Third diagonal of A027447.

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A027461 Third column of A027447.

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A027448 Triangle read by rows: 4th power of the lower triangular mean matrix (M[i,j] = 1/i for i <= j).

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A027446 Triangle read by rows: square of the lower triangular mean matrix.

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A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.

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