A047918 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d) if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n).

1, 2, 0, 6, 0, 0, 8, 0, 0, 16, 20, 0, 0, 0, 100, 12, 24, 36, 0, 0, 648, 42, 0, 0, 0, 0, 0, 4998, 32, 32, 0, 320, 0, 0, 0, 39936, 54, 0, 270, 0, 0, 0, 0, 0, 362556, 40, 160, 0, 0, 3800, 0, 0, 0, 0, 3624800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916690, 48, 96

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice
• easy

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

Cf. A008683, A027750, A225817.

Programs

Haskell

a047918 n k = sum [a008683 (fromIntegral d) * a047916 n (k `div` d) |
                   mod n k == 0, d <- [1..k], mod k d == 0]
a047918_row n = map (a047918 n) [1..n]
a047918_tabl = map a047918_row [1..]
-- Reinhard Zumkeller, Mar 19 2014

Mathematica

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

Extensions

Offset corrected by Reinhard Zumkeller, Mar 19 2014