A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

1, 2, 1, 3, 0, 2, 4, 2, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 4, 0, 4, 0, 0, 0, 4, 9, 0, 6, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 6, 8, 6, 0, 4, 0, 0, 0, 0, 0, 4, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 14, 7, 0, 0, 0, 0, 12, 0, 0, 0

Offset

1,2

Comments

If the two matrices A054523 and A054522 are commuted, the matrix product becomes A127477.

Links

Table of n, a(n) for n=1..101.

Formula

T(n,k) = sum_{j=k..n} A054523(n,j) * A054522(j,k).

T(n,n) = A000010(n) (diagonal).

sum_{k=1..n} T(n,k) = A018804(n) (row sums).

Example

First few rows of the triangle are:
.1;
.2, 1;
.3, 0, 2;
.4, 2, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 4, 0, 4, 0, 0, 0, 4;
....

Maple

A054522 := proc(n, k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n, k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127478 := proc(n, k) add( A054523(n, j)*A054522(j, k), j=k..n) ; end: seq(seq( A127478(n, k), k=1..n), n=1..15) ;

Crossrefs

Cf. A054522, A054523, A018804, A000010.

Sequence in context: A025649 A025642 A025643 * A127472 A194665 A004563

Adjacent sequences: A127475 A127476 A127477 * A127479 A127480 A127481

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jan 15 2007

Extensions

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

Status

approved