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

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

Author

Olivier Gérard

Keywords

• nonn
• tabl

Comments

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]

Examples

			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
		

Links

• L. Bendersky, Sur la fonction gamma généralisée, Acta Math. 61 (1933), p. 263-322. See p. 295.

Crossrefs

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.

Programs

Mathematica

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 *)

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

Formula

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.

Extensions

Edited by M. F. Hasler, Nov 05 2019