|
%I #30 Sep 08 2022 08:45:47
%S 9,64,25,225,144,49,576,441,256,81,1225,1024,729,400,121,2304,2025,
%T 1600,1089,576,169,3969,3600,3025,2304,1521,784,225,6400,5929,5184,
%U 4225,3136,2025,1024,289,9801,9216,8281,7056,5625,4096,2601,1296,361,14400,13689,12544,11025,9216,7225,5184,3249,1600,441,20449,19600,18225,16384,14161,11664,9025,6400,3969,1936,529
%N T(n,k) = n^4 - 2*k^2*n^2 + k^4 = A120070(n, k)^2.
%H G. C. Greubel, <a href="/A165447/b165447.txt">Rows n=2..100 of triangle, flattened</a>
%F a(n) = (R*(R+3)-2*(n-3))^2*(R*(1-R)+2*(n+1))^2/16 where R = floor((sqrt(8*n-15)-1)/2). - _Luce ETIENNE_, Jun 04 2017
%F G.f.: (x*(1 + 11*x + 11*x^2 + x^3)*(-1 + y)^4 - 2*(-1 + x)^2*x*(1 + x)*(-1 + y)^2*y*(1 + y) + (-1 + x)^4*y*(1 + 11*y + 11*y^2 + y^3))/((-1 + x)^5*(-1 + y)^5). - _Stefano Spezia_, Oct 21 2018
%e Triangle begins:
%e 9;
%e 64, 25;
%e 225, 144, 49;
%e 576, 441, 256, 81;
%e 1225, 1024, 729, 400, 121;
%e 2304, 2025, 1600, 1089, 576, 169;
%e ...
%p a:=(n,k)->(n^2-k^2)^2: seq(seq(a(n,k),k=1..n-1),n=2..12); # _Muniru A Asiru_, Oct 21 2018
%t Table[n^2 - k^2, {n, 2, 12}, {k, n - 1}]^2 // Flatten (* _Michael De Vlieger_, Jun 06 2017, after _Alonso del Arte_ at A120070 *)
%t f[n_] := Table[SeriesCoefficient[(x (1 + 11 x + 11 x^2 + x^3) (-1 + y)^4 - 2 (-1 + x)^2 x (1 + x) (-1 + y)^2 y (1 + y) + (-1 + x)^4 y (1 + 11 y + 11 y^2 + y^3))/((-1 + x)^5 (-1 + y)^5) , {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 1, n-1}]; Flatten[Array[f, 11]] (* _Stefano Spezia_, Oct 21 2018 *)
%o (PARI) for (n=2, 10, for(k=1, n-1, print1((n^2-k^2)^2, ", ")); print()); \\ _Michel Marcus_, Jun 04 2017
%o (GAP) Flat(List([2..12],n->List([1..n-1],k->n^4-2*k^2*n^2+k^4))); # _Muniru A Asiru_, Oct 21 2018
%o (Magma) [(Floor((Sqrt(8*n-15)-1)/2)*(Floor((Sqrt(8*n-15)-1)/2)+3)-2*(n-3))^2*(Floor((Sqrt(8*n-15)-1)/2)*(1-Floor((Sqrt(8*n-15)-1)/2))+2*(n+1))^2/16: n in [2..30]]; // _G. C. Greubel_, Oct 20 2018
%Y Cf. A120070.
%K nonn,tabl,easy
%O 2,1
%A _Paul Curtz_, Sep 20 2009
%E More terms from _Michel Marcus_, Jun 04 2017
|
