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

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

1,2

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A049375 A049224 A223517 * A027518 A027539 A027479

Adjacent sequences: A027445 A027446 A027447 * A027449 A027450 A027451

Keyword

nonn,tabl

Author

Olivier Gérard

Extensions

Edited by Robert Israel, Oct 05 2019

Status

approved