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A082110 Array A(n,k) = (k*n)^2 + 5*(k*n) + 1, read by antidiagonals.

%I A082110 #11 Dec 23 2022 07:52:09
%S A082110 1,1,1,1,7,1,1,15,15,1,1,25,37,25,1,1,37,67,67,37,1,1,51,105,127,105,
%T A082110 51,1,1,67,151,205,205,151,67,1,1,85,205,301,337,301,205,85,1,1,105,
%U A082110 267,415,501,501,415,267,105,1,1,127,337,547,697,751,697,547,337,127,1
%N A082110 Array A(n,k) = (k*n)^2 + 5*(k*n) + 1, read by antidiagonals.
%H A082110 G. C. Greubel, <a href="/A082110/b082110.txt">Antidiagonals n = 0..50, flattened</a>
%F A082110 A(n, k) = (k*n)^2 + 5*(k*n) + 1 (Square array).
%F A082110 A(k, n) = A(n, k).
%F A082110 A(2, k) = A082111(k).
%F A082110 A(3, k) = A082112(k).
%F A082110 A(n, n) = T(2*n, n) = A082113(n) (main diagonal).
%F A082110 T(n, k) = (k*(n-k))^2 + 5*k*(n-k) + 1 (number triangle).
%F A082110 Sum_{k=0..n} T(n, k) = A082114(n) (diagonal sums of the array).
%F A082110 From _G. C. Greubel_, Dec 22 2022: (Start)
%F A082110 T(n, n-k) = T(n, k).
%F A082110 Sum_{k=0..n} (-1)^k*T(n, k) = (1 - 3*n)*(1 + (-1)^n)/2. (End)
%e A082110 Square array, A(n, k), begins as:
%e A082110   1,   1,   1,   1,    1,    1,    1,    1,    1, ... A000012;
%e A082110   1,   7,  15,  25,   37,   51,   67,   85,  105, ... A082111;
%e A082110   1,  15,  37,  67,  105,  151,  205,  267,  337, ... A082112;
%e A082110   1,  25,  67, 127,  205,  301,  415,  547,  697, ...
%e A082110   1,  37, 105, 205,  337,  501,  697,  925, 1185, ...
%e A082110   1,  51, 151, 301,  501,  751, 1051, 1401, 1801, ...
%e A082110   1,  67, 205, 415,  697, 1051, 1477, 1975, 2545, ...
%e A082110   1,  85, 267, 547,  925, 1401, 1975, 2647, 3417, ...
%e A082110   1, 105, 337, 697, 1185, 1801, 2545, 3417, 4417, ...
%e A082110 Antidiagonals, T(n, k), begins as:
%e A082110   1;
%e A082110   1,  1;
%e A082110   1,  7,   1;
%e A082110   1, 15,  15,   1;
%e A082110   1, 25,  37,  25,   1;
%e A082110   1, 37,  67,  67,  37,   1;
%e A082110   1, 51, 105, 127, 105,  51,   1;
%e A082110   1, 67, 151, 205, 205, 151,  67,  1;
%e A082110   1, 85, 205, 301, 337, 301, 205, 85,  1;
%t A082110 T[n_, k_]:= (k*(n-k))^2 + 5*(k*(n-k)) + 1;
%t A082110 Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2022 *)
%o A082110 (Magma) [(k*(n-k))^2 + 5*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // _G. C. Greubel_, Dec 22 2022
%o A082110 (SageMath)
%o A082110 def A082110(n,k): return (k*(n-k))^2 + 5*(k*(n-k)) + 1
%o A082110 flatten([[A082110(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 22 2022
%Y A082110 Cf. A082039, A082043, A082046, A082105, A082111, A082112, A082113, A082114.
%K A082110 easy,nonn,tabl
%O A082110 0,5
%A A082110 _Paul Barry_, Apr 04 2003