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A127477
Triangle T(n,k) read by rows: matrix product A054522 * A054523.
2
1, 2, 1, 5, 0, 2, 6, 3, 0, 2, 17, 0, 0, 0, 4, 10, 5, 4, 0, 0, 2, 37, 0, 0, 0, 0, 0, 6, 22, 11, 0, 6, 0, 0, 0, 4, 41, 0, 14, 0, 0, 0, 0, 0, 6, 34, 17, 0, 0, 8, 0, 0, 0, 0, 4, 101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 30, 15, 12, 10, 0, 6, 0, 0, 0, 0, 0, 4, 145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 74, 37, 0
OFFSET
1,2
COMMENTS
If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.
FORMULA
T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
sum_{k=1..n} T(n,k) = A057660(n) (row sums).
T(n,n) = A000010(n) (diagonal).
T(n,1) = A029939(n).
EXAMPLE
First few rows of the triangle are:
1;
2, 1;
5, 0, 2;
6, 3, 0, 2;
17, 0, 0, 0, 4;
10, 5, 4, 0, 0, 2;
37, 0, 0, 0, 0, 0, 6;
22, 11, 0, 6, 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:
A127477 := proc(n, k) add( A054522(n, j)*A054523(j, k), j=k..n) ; end: seq(seq( A127477(n, k), k=1..n), n=1..15) ;
CROSSREFS
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