A055894 - OEIS

%I #21 Sep 04 2022 22:21:41

%S 1,1,1,2,2,2,2,3,3,2,3,4,8,4,3,2,5,10,10,5,2,4,6,18,22,18,6,4,2,7,21,

%T 35,35,21,7,2,4,8,32,56,78,56,32,8,4,3,9,36,87,126,126,87,36,9,3,4,10,

%U 50,120,220,254,220,120,50,10,4,2,11,55,165,330,462,462,330,165,55,11

%N Inverse Moebius transform of Pascal's triangle A007318.

%H G. C. Greubel, <a href="/A055894/b055894.txt">Table of n, a(n) for the first 101 rows, flattened</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%e Triangle starts:

%e   [0]  1;

%e   [1]  1,   1;

%e   [2]  2,   2,   2;

%e   [3]  2,   3,   3,   2;

%e   [4]  3,   4,   8,   4,   3;

%e   [5]  2,   5,  10,  10,   5,   2;

%e   [6]  4,   6,  18,  22,  18,   6,   4;

%e   [7]  2,   7,  21,  35,  35,  21,   7,   2;

%e   [8]  4,   8,  32,  56,  78,  56,  32,   8,   4;

%e   [9]  3,   9,  36,  87, 126, 126,  87,  36,   9,   3;

%e   ...

%t T[n_, k_] := DivisorSum[GCD[k, n], Binomial[n/#, k/#] &]; T[0, 0] = 1; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 02 2015 *)

%o (PARI)

%o T(n,k) = if(n<=0, n==0, sumdiv(gcd(n,k), d, binomial(n/d,k/d) ) );

%o /* print triangle: */

%o { for (n=0, 17, for (k=0, n, print1(T(n,k),", "); ); print(); ); }

%o /* _Joerg Arndt_, Oct 21 2012 */

%Y Cf. A007318.

%Y Row sums give A055895.

%K nonn,tabl

%O 0,4

%A _Christian G. Bower_, Jun 09 2000