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A055894 Inverse Moebius transform of Pascal's triangle A007318.

%I A055894 #21 Sep 04 2022 22:21:41
%S A055894 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 A055894 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 A055894 50,120,220,254,220,120,50,10,4,2,11,55,165,330,462,462,330,165,55,11
%N A055894 Inverse Moebius transform of Pascal's triangle A007318.
%H A055894 G. C. Greubel, <a href="/A055894/b055894.txt">Table of n, a(n) for the first 101 rows, flattened</a>
%H A055894 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A055894 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A055894 Triangle starts:
%e A055894   [0]  1;
%e A055894   [1]  1,   1;
%e A055894   [2]  2,   2,   2;
%e A055894   [3]  2,   3,   3,   2;
%e A055894   [4]  3,   4,   8,   4,   3;
%e A055894   [5]  2,   5,  10,  10,   5,   2;
%e A055894   [6]  4,   6,  18,  22,  18,   6,   4;
%e A055894   [7]  2,   7,  21,  35,  35,  21,   7,   2;
%e A055894   [8]  4,   8,  32,  56,  78,  56,  32,   8,   4;
%e A055894   [9]  3,   9,  36,  87, 126, 126,  87,  36,   9,   3;
%e A055894   ...
%t A055894 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 A055894 (PARI)
%o A055894 T(n,k) = if(n<=0, n==0, sumdiv(gcd(n,k), d, binomial(n/d,k/d) ) );
%o A055894 /* print triangle: */
%o A055894 { for (n=0, 17, for (k=0, n, print1(T(n,k),", "); ); print(); ); }
%o A055894 /* _Joerg Arndt_, Oct 21 2012 */
%Y A055894 Cf. A007318.
%Y A055894 Row sums give A055895.
%K A055894 nonn,tabl
%O A055894 0,4
%A A055894 _Christian G. Bower_, Jun 09 2000