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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).

%I A027448 #22 Nov 06 2019 18:20:44
%S A027448 1,15,1,575,65,8,5845,865,175,27,874853,153713,39743,9963,1728,
%T A027448 1009743,200403,60333,19153,5368,1000,389919909,84873489,28400079,
%U A027448 10419739,3681784,1105000,216000,3449575767,807843807,292420227
%N A027448 Triangle read by rows: 4th power of the lower triangular mean matrix (M[i,j] = 1/i for i <= j).
%H A027448 Robert Israel, <a href="/A027448/b027448.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%F A027448 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
%F A027448 That is, the fractions in M^4 are written using the least common denominator before taking the numerators. - _M. F. Hasler_, Nov 05 2019
%e A027448 Table starts:
%e A027448           1
%e A027448          15           1
%e A027448         575          65           8
%e A027448        5845         865         175          27
%e A027448      874853      153713       39743        9963        1728
%e A027448     1009743      200403       60333       19153        5368        1000
%p A027448 Rows:= 10:
%p A027448 M:= Matrix(Rows,Rows,(i,j) -> `if`(i>=j,1/i,0)):
%p A027448 B:= M^4:
%p A027448 L:= [seq(ilcm(seq(denom(B[i,j]),j=1..i)),i=1..Rows)]:
%p A027448 seq(seq(B[i,j]*L[i],j=1..i),i=1..Rows); # _Robert Israel_, Oct 05 2019
%t A027448 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 *)
%o A027448 (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
%Y A027448 Cf. A027446 (square of M), A027447 (cube of M).
%K A027448 nonn,tabl
%O A027448 1,2
%A A027448 _Olivier Gérard_
%E A027448 Edited by _Robert Israel_, Oct 05 2019