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%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
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