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

1,2

Comments

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

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]

Example

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

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 *)

Prog

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

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.

Sequence in context: A274440 A378208 A199220 * A101207 A186435 A260297

Adjacent sequences: A047913 A047914 A047915 * A047917 A047918 A047919

Keyword

nonn,tabl,nice,easy

Author

N. J. A. Sloane

Status

approved