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A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n). 2

%I

%S 1,1,0,2,0,0,2,0,0,4,4,0,0,0,20,2,4,6,0,0,108,6,0,0,0,0,0,714,4,4,0,

%T 40,0,0,0,4992,6,0,30,0,0,0,0,0,40284,4,16,0,0,380,0,0,0,0,362480,10,

%U 0,0,0,0,0,0,0,0,0,3628790,4,8,60,312,0,3768,0,0,0,0

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

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

%H Reinhard Zumkeller, <a href="/A047919/b047919.txt">Rows n = 1..125 of triangle, flattened</a>

%H C. L. Mallows and N. J. A. Sloane, <a href="/A002618/a002618_1.pdf">Notes on A002618, A002619, etc.</a>

%H N. J. A. Sloane, <a href="/A002618/a002618_2.pdf">Notes on A002618, A002619, etc.</a>

%H J. E. A. Steggall, <a href="http://www.handweaving.net/DAItemDetail.aspx?ItemID=3237">On the numbers of patterns which can be derived from certain elements</a>, Mess. Math., 37 (1907), 56-61.

%H J. E. A. Steggall, <a href="/A002618/a002618.pdf">On the numbers of patterns which can be derived from certain elements</a>, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

%t 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]}]; row[n_] := Table[a[n, k], {k, 1, n}]/n; Table[row[n], {n, 1, 12}] // Flatten (* _Jean-Fran├žois Alcover_, Nov 21 2012, after A047918 *)

%o (Haskell)

%o a047919 n k = a047919_tabl !! (n-1) !! (k-1)

%o a047919_row n = a047919_tabl !! (n-1)

%o a047919_tabl = zipWith (zipWith div) a047918_tabl a002024_tabl

%o -- _Reinhard Zumkeller_, Mar 19 2014

%Y Divide n-th row of array A047918 by n.

%Y Cf. A002024.

%K nonn,tabl,nice,easy

%O 1,4

%A _N. J. A. Sloane_

%E Offset corrected by _Reinhard Zumkeller_, Mar 19 2014

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Last modified April 8 00:29 EDT 2020. Contains 333311 sequences. (Running on oeis4.)