A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

Comments

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

Examples

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...
		

References

• J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
• N. J. A. Sloane, Notes on A002618, A002619, etc.
• J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
• J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

Crossrefs

A064649 gives the row sums.

Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).

See also A247917, A047918, A047919.

Programs

Haskell

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

Mathematica

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

PARI

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017