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

%I A027446 #30 May 06 2022 17:00:42
%S A027446 1,3,1,11,5,2,25,13,7,3,137,77,47,27,12,147,87,57,37,22,10,1089,669,
%T A027446 459,319,214,130,60,2283,1443,1023,743,533,365,225,105,7129,4609,3349,
%U A027446 2509,1879,1375,955,595,280,7381,4861,3601,2761,2131,1627,1207,847,532,252
%N A027446 Triangle read by rows: square of the lower triangular mean matrix.
%C A027446 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]
%H A027446 L. Bendersky, <a href="https://projecteuclid.org/journals/acta-mathematica/volume-61/issue-none/Sur-la-fonction-gamma-g%C3%A9n%C3%A9ralis%C3%A9e/10.1007/BF02547794.full">Sur la fonction gamma généralisée</a>, Acta Math. 61 (1933), p. 263-322. See p. 295.
%F A027446 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).
%F A027446 a(i,j) = lcm(seq(A119948(i,m),m=1..i))*A119947(i,j)/A119948(i,j), 1 <= j =< i and zero otherwise.
%e A027446 Triangle starts
%e A027446      1
%e A027446      3,    1
%e A027446     11,    5,    2
%e A027446     25,   13,    7,    3
%e A027446    137,   77,   47,   27,   12
%e A027446    147,   87,   57,   37,   22,   10
%e A027446   1089,  669,  459,  319,  214,  130,  60
%e A027446   2283, 1443, 1023,  743,  533,  365, 225, 105
%e A027446   7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280
%e A027446   ... - _Joerg Arndt_, Mar 29 2013
%t A027446 rows = 10;
%t A027446 M = MatrixPower[Table[If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}], 2];
%t A027446 T = Table[M[[n]]*LCM @@ Denominator[M[[n]]], {n, 1, rows}];
%t A027446 Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 05 2013, updated May 06 2022 *)
%o A027446 (PARI) 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
%Y A027446 The row sums give A081528(n), n>=1.
%Y A027446 The column sequences give A025529, A027457, A027458 for j=1..3.
%Y A027446 The diagonal sequences give A002944, A027449, A027450.
%Y A027446 Cf. A027447, A027448.
%K A027446 nonn,tabl
%O A027446 1,2
%A A027446 _Olivier Gérard_
%E A027446 Edited by _M. F. Hasler_, Nov 05 2019