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A098172 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

%I A098172 #22 Sep 08 2022 08:45:14
%S A098172 1,0,1,0,0,1,0,0,1,1,0,0,0,4,1,0,0,0,0,10,1,0,0,0,0,1,20,1,0,0,0,0,0,
%T A098172 7,35,1,0,0,0,0,0,0,28,56,1,0,0,0,0,0,0,1,84,84,1,0,0,0,0,0,0,0,10,
%U A098172 210,120,1,0,0,0,0,0,0,0,0,55,462,165,1,0,0,0,0,0,0,0,0,1,220,924,220,1
%N A098172 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
%C A098172 Row sums are A024493.
%C A098172 From _R. J. Mathar_, Mar 22 2013: (Start)
%C A098172 The matrix inverse starts
%C A098172   1;
%C A098172   0, 1;
%C A098172   0, 0,      1;
%C A098172   0, 0,     -1,     1;
%C A098172   0, 0,      4,    -4,     1;
%C A098172   0, 0,    -40,    40,   -10,   1;
%C A098172   0, 0,    796,  -796,   199, -20,   1;
%C A098172   0, 0, -27580, 27580, -6895, 693, -35, 1;
%C A098172   ... (End)
%H A098172 Seiichi Manyama, <a href="/A098172/b098172.txt">Rows n = 0..139, flattened</a>
%F A098172 Triangle T(n, k) = binomial(n, 3(n-k)).
%e A098172 Rows begin
%e A098172   {1},
%e A098172   {0,1},
%e A098172   {0,0,1},
%e A098172   {0,0,1,1},
%e A098172   {0,0,0,4,1},
%e A098172   {0,0,0,0,10,1},
%e A098172   ...
%t A098172 Table[Binomial[n, 3(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 15 2019 *)
%o A098172 (PARI) {T(n, k) = binomial(n, 3*(n-k))}; \\ _G. C. Greubel_, Mar 15 2019
%o A098172 (Magma) [[Binomial(n, 3*(n-k)): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Mar 15 2019
%o A098172 (Sage) [[binomial(n, 3*(n-k)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 15 2019
%o A098172 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 3*(n-k)) ))); # _G. C. Greubel_, Mar 15 2019
%Y A098172 Cf. A098158.
%K A098172 easy,nonn,tabl
%O A098172 0,14
%A A098172 _Paul Barry_, Aug 30 2004