A098172 - OEIS

%I #22 Sep 08 2022 08:45:14

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

%C Row sums are A024493.

%C From _R. J. Mathar_, Mar 22 2013: (Start)

%C The matrix inverse starts

%C   1;

%C   0, 1;

%C   0, 0,      1;

%C   0, 0,     -1,     1;

%C   0, 0,      4,    -4,     1;

%C   0, 0,    -40,    40,   -10,   1;

%C   0, 0,    796,  -796,   199, -20,   1;

%C   0, 0, -27580, 27580, -6895, 693, -35, 1;

%C   ... (End)

%H Seiichi Manyama, <a href="/A098172/b098172.txt">Rows n = 0..139, flattened</a>

%F Triangle T(n, k) = binomial(n, 3(n-k)).

%e Rows begin

%e   {1},

%e   {0,1},

%e   {0,0,1},

%e   {0,0,1,1},

%e   {0,0,0,4,1},

%e   {0,0,0,0,10,1},

%e   ...

%t Table[Binomial[n, 3(n-k)], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 15 2019 *)

%o (PARI) {T(n, k) = binomial(n, 3*(n-k))}; \\ _G. C. Greubel_, Mar 15 2019

%o (Magma) [[Binomial(n, 3*(n-k)): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Mar 15 2019

%o (Sage) [[binomial(n, 3*(n-k)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 15 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 3*(n-k)) ))); # _G. C. Greubel_, Mar 15 2019

%Y Cf. A098158.

%K easy,nonn,tabl

%O 0,14

%A _Paul Barry_, Aug 30 2004