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A027446 Triangle read by rows: square of the lower triangular mean matrix. 10

%I #30 May 06 2022 17:00:42

%S 1,3,1,11,5,2,25,13,7,3,137,77,47,27,12,147,87,57,37,22,10,1089,669,

%T 459,319,214,130,60,2283,1443,1023,743,533,365,225,105,7129,4609,3349,

%U 2509,1879,1375,955,595,280,7381,4861,3601,2761,2131,1627,1207,847,532,252

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

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

%e Triangle starts

%e 1

%e 3, 1

%e 11, 5, 2

%e 25, 13, 7, 3

%e 137, 77, 47, 27, 12

%e 147, 87, 57, 37, 22, 10

%e 1089, 669, 459, 319, 214, 130, 60

%e 2283, 1443, 1023, 743, 533, 365, 225, 105

%e 7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280

%e ... - _Joerg Arndt_, Mar 29 2013

%t rows = 10;

%t M = MatrixPower[Table[If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}], 2];

%t T = Table[M[[n]]*LCM @@ Denominator[M[[n]]], {n, 1, rows}];

%t Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 05 2013, updated May 06 2022 *)

%o (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 The row sums give A081528(n), n>=1.

%Y The column sequences give A025529, A027457, A027458 for j=1..3.

%Y The diagonal sequences give A002944, A027449, A027450.

%Y Cf. A027447, A027448.

%K nonn,tabl

%O 1,2

%A _Olivier Gérard_

%E Edited by _M. F. Hasler_, Nov 05 2019

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)