A027448 -id:A027448 - cp's OEIS frontend

A027463 Second column of A027448.

1, 65, 865, 153713, 200403, 84873489, 807843807, 25265669869, 28682861185, 42667254975995, 47094524452625, 113048851861209965, 122484165188229545, 131772609157313825, 1127325729624166675, 5892299098094866152275

Offset: 2

Olivier Gérard

nonn

Numerators of sequence a[ 2, n ] in (a[ i, j ])^4 where a[ i, j ] = 1/i if j<=i, 0 if j>i

A329122 First column of A027448.

Original entry on oeis.org

1, 15, 575, 5845, 874853, 1009743, 389919909, 3449575767, 101622655189, 109727076745, 156384687564755, 166320156938585, 386435095871809685, 406739013616700465, 426379471078085225, 3563283104558049475, 18232291817356124003075, 18937687901195398630875

Offset: 1

Sean A. Irvine, Nov 05 2019

nonn

This sequences was originally A027462, but differs from the current version of A027462 for n >= 6. This difference arises because the rational numbers in the matrix defining A027448 (and hence this sequence) are not reduced to the lowest common denominator, whereas the values in A027462 are.

Cf. A027448, A027462, A027463.

A027454 First diagonal of A027448.

Original entry on oeis.org

1, 1, 8, 27, 1728, 1000, 216000, 1157625, 21952000, 16003008, 16003008000, 12326391000, 21300003648000, 17053975224000, 13865513485824, 91398648466125, 374368864117248000, 315376239610432000

Offset: 1

Olivier Gérard

nonn

Numerators of sequence a[ n, n ] in (a[ i, j ])^4 where a[ i, j ] = 1/i if j<=i, 0 if j>i

A027455 Second diagonal of A027448.

Original entry on oeis.org

15, 65, 175, 9963, 5368, 1105000, 5720625, 105686875, 75492928, 74269960128, 56445795000, 96454009575000, 76498373160000, 61692439558464, 403805999193597, 1643804692663258125, 1377243480733120000

Offset: 2

Olivier Gérard

nonn

Numerators of sequence a[ n, n-1 ] in (a[ i, j ])^4 where a[ i, j ] = 1/i if j<=i, 0 if j>i

A027456 Third diagonal of A027448.

Original entry on oeis.org

575, 865, 39743, 19153, 3681784, 18198125, 325151875, 226462663, 218431158848, 163390785768, 275577122835000, 216187828065000, 172736941569984, 1121674137386517, 4534605234349771581, 3776294388104363125, 21799582086436073920000, 18520810592837638076992

Offset: 3

Olivier Gérard

nonn

Cf. A027448.

Numerators of sequence a[ n, n-2 ] in (a[ i, j ])^4 where a[ i, j ] = 1/i if j<=i, 0 if j>i.

More terms from Sean A. Irvine, Nov 04 2019

A027464 Third column of A027448.

Original entry on oeis.org

8, 175, 39743, 60333, 28400079, 292420227, 9723178009, 11596904605, 17970237912815, 20530907592845, 50764671209056505, 56436971899092485, 62109279389181725, 542166878698918075, 2885442896258516289275, 3106224746146152726675, 22811796735204826462092225

Offset: 3

Olivier Gérard

nonn

Cf. A027448.

Numerators of sequence a[ 3, n ] in (a[ i, j ])^4 where a[ i, j ] = 1/i if j<=i, 0 if j>i.

More terms from Sean A. Irvine, Nov 05 2019

A027447 Triangle read by rows: cube of the lower triangular mean matrix.

Original entry on oeis.org

1, 7, 1, 85, 19, 4, 415, 115, 37, 9, 12019, 3799, 1489, 549, 144, 13489, 4669, 2059, 919, 364, 100, 726301, 268921, 128431, 64171, 30676, 12700, 3600, 3144919, 1227199, 621139, 334699, 178669, 89125, 38025, 11025, 30300391, 12335311, 6527971, 3714811, 2134141, 1187125, 609625, 265825, 78400

Offset: 1

Olivier Gérard

			Triangle begins:
      1;
      7,    1;
     85,   19,    4;
    415,  115,   37,   9;
  12019, 3799, 1489, 549, 144,
  ...

Cf. A027446, A027448.

rows = 9; m = Table[ If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}]; m3 = m.m.m; Table[ fracs = m3[[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 *)

tabl(nn) = {my(M = matrix(nn, nn, i, j, if (j<=i, 1/i, 0))^3); for (n=1, nn, my(row = M[n,1..n]); print(denominator(row)*row))} \\ Michel Marcus, Nov 05 2019, edited by M. F. Hasler, Nov 05 2019

A027447_row(n)=denominator(n=(matrix(n,n, i,j, (j<=i)/i)^3)[n,])*n \\ M. F. Hasler, Nov 05 2019

Let A be the matrix with A[i,j] = 1/i if j <= i, 0 if j > i. Then this table lists the numerators in A^3 when each row is written using the least common denominator. [Edited by M. F. Hasler, Nov 05 2019]

More terms from Michel Marcus, Nov 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

A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

Original entry on oeis.org

1, 3, 1, 11, 5, 1, 25, 13, 7, 1, 137, 77, 47, 9, 1, 49, 29, 19, 37, 11, 1, 363, 223, 153, 319, 107, 13, 1, 761, 481, 341, 743, 533, 73, 15, 1, 7129, 4609, 3349, 2509, 1879, 275, 191, 17, 1, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 121, 19, 1, 83711, 55991, 42131, 32891, 25961

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of the corresponding denominators is A119948. The rationals appear in lowest terms (while in A027446 they are row-wise on the least common denominator).
The triangle with row number i multiplied with the least common multiple (LCM) of its denominators yields A027446.
First column is A001008. - Tilman Neumann, Oct 01 2008
Column 2 is A064169. - Clark Kimberling, Aug 13 2012
Third diagonal (11, 13, 47, ...) is A188386. - Clark Kimberling, Aug 13 2012

			The rationals are [1]; [3/4, 1/4]; [11/18, 5/18, 1/9]; [25/48, 13/48, 7/48, 1/16]; ... See the W. Lang link for more.
From _Clark Kimberling_, Aug 13 2012: (Start)
As a triangle given by f(n,m) = Sum_{h=m..n} 1/h, the first six rows are:
    1
    3    1
   11    5    1
   25   13    7    1
  137   77   47    9    1
   49   29   19   37   11    1
  363  223  153  319  107   13    1
(End)

Wolfdieter Lang, First ten rows and rationals.

Cf. A002024: i appears i times (denominators in row i of the matrix A).
Row sums give A119949. Row sums of the triangle of rationals always give 1.
For the cube of this matrix see the rational triangle A119935/A119932 and A027447; see A027448 for the fourth power.
Cf. A001008, A027446, A064169, A119948, A188386.

f[n_, m_] := Numerator[Sum[1/k, {k, m, n}]]
Flatten[Table[f[n, m], {n, 1, 10}, {m, 1, n}]]
TableForm[Table[f[n, m], {n, 1, 10}, {m, 1, n}]] (* Clark Kimberling, Aug 13 2012 *)

A119947_upto(n)={my(M=matrix(n,n,i,j,(j<=i)/i)^2);vector(n,r,apply(numerator,M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

a(i,j) = numerator(r(i,j)) with r(i,j):=(A^2)[i,j], where the matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i, (lower triangular).

Edited by M. F. Hasler, Nov 05 2019

A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).

Original entry on oeis.org

1, 15, 575, 5845, 874853, 336581, 129973303, 1149858589, 101622655189, 21945415349, 31276937512951, 33264031387717, 77287019174361937, 81347802723340093, 17055178843123409, 142531324182321979

Offset: 1

Olivier Gérard

nonn

Originally defined as the first column of A027448, but now contains numerator in reduced form (cf. A329122). - Sean A. Irvine, Nov 05 2019

Donal F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Vol. II(a), arXiv:0710.4023 [math.HO], 2007-2008.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

Cf. A027459.

a[n_] := -1/6 Integrate[x^(n-1) Log[1-x]^3, {x, 0, 1}] // Numerator;
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Aug 06 2018 *)

Numerators of sequence a(1,n) in (a(i,j))^4 where a(i,j) = 1/i if j <= i, 0 if j > i.
Numerators of (H(n,1)^3 + 3*H(n,1)*H(n,2) + 2*H(n,3))/(6*n) = ((gamma+Psi(n+1))^3 + 3*(gamma+Psi(n+1))*(1/6*Pi^2 - Psi(1, n+1)) + 2*Zeta(3) + Psi(2, n+1))/(6*n), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers. - Vladeta Jovovic, Aug 10 2002
From Groux Roland, Feb 11 2011: (Start)
For n>=1, (H(n,1)^3 + 3*H(n,1)*H(n,2) + 2*H(n,3))/(6*n) = -(1/6)*Integral_{x=0..1} x^(n-1)*log^3(1-x) dx = (1/n)*Sum_{j=1..n} (H(n,1) - H(j-1,1))*H(j,1)/j.
For every k>=1 the first column of (a(i,j))^k is the binomial transform of (-1)^n/(n+1)^k.
Proof: the sequence S(n,k) = ((-1)^k/k!)*Integral_{x=0..1} x^(n-1)*log^k(1-x) dx gives the binomial transform of (-1)^n/(n+1)^(k+1) and can be evaluated by parts with S(n,k) = (1/n)*Sum_{j=1..k} S(j,k-1) according to the generation of the first column of (a(i,j))^k.
(End)

Corrected by Vladeta Jovovic, Aug 10 2002

Title changed by Sean A. Irvine, Nov 05 2019

cp's OEIS Frontend

A027463 Second column of A027448.

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A329122 First column of A027448.

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A027454 First diagonal of A027448.

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A027455 Second diagonal of A027448.

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A027456 Third diagonal of A027448.

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A027464 Third column of A027448.

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A027447 Triangle read by rows: cube of the lower triangular mean matrix.

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

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A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

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A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).

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