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

1,2

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]

Links

Table of n, a(n) for n=1..55.

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

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.

Example

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

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 *)

Prog

(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

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.

Sequence in context: A099001 A119947 A165674 * A027516 A092808 A343171

Adjacent sequences: A027443 A027444 A027445 * A027447 A027448 A027449

Keyword

nonn,tabl

Author

Olivier Gérard

Extensions

Edited by M. F. Hasler, Nov 05 2019

Status

approved