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A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

%I A050186 #35 Jul 16 2022 11:43:51
%S A050186 1,1,1,0,2,0,0,3,3,0,0,4,4,4,0,0,5,10,10,5,0,0,6,12,18,12,6,0,0,7,21,
%T A050186 35,35,21,7,0,0,8,24,56,64,56,24,8,0,0,9,36,81,126,126,81,36,9,0,0,10,
%U A050186 40,120,200,250,200,120,40,10,0,0,11,55,165,330,462,462,330,165,55,11
%N A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.
%H A050186 M. F. Hasler, <a href="/A050186/a050186_1.txt">A050186, rows 0..50</a>, Sep 27 2018
%H A050186 M. F. Hasler, <a href="/A050186/a050186_2.txt">A050186, rows 0..100</a>, Sep 27 2018
%H A050186 M. F. Hasler, <a href="/A050186/a050186.txt">A050186, rows 0..200</a>, Sep 27 2018
%H A050186 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A050186 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A050186 MOEBIUS transform of A007318 Pascal's Triangle.
%F A050186 If rows n > 1 are divided by n, this yields the triangle A051168, which equals A245558 surrounded by 0's (except for initial terms). This differs from A011847 from row n = 9 on. - _M. F. Hasler_, Sep 29 2018
%e A050186 For example, T(4,2) counts 1100,1001,0011,0110; T(2,1) counts 10, 01 (hence also counts 1010, 0101).
%e A050186 Rows:
%e A050186   1;
%e A050186   1,  1;
%e A050186   0,  2,  0;
%e A050186   0,  3,  3,  0;
%e A050186   0,  4,  4,  4,  0;
%e A050186   0,  5, 10, 10,  5,  0;
%t A050186 T[n_, k_] := If[n == 0, 1, DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#]&]];
%t A050186 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 16 2022 *)
%o A050186 (PARI) A050186(n,k)=sumdiv(gcd(n+!n,k),d,moebius(d)*binomial(n/d,k/d)) \\ _M. F. Hasler_, Sep 27 2018
%Y A050186 Same triangle as A053727 except this one includes column 0.
%Y A050186 T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.
%K A050186 nonn,tabl,nice
%O A050186 0,5
%A A050186 _Clark Kimberling_