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A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.

%I A082105 #14 Dec 24 2022 03:52:51
%S A082105 1,1,1,1,6,1,1,13,13,1,1,22,33,22,1,1,33,61,61,33,1,1,46,97,118,97,46,
%T A082105 1,1,61,141,193,193,141,61,1,1,78,193,286,321,286,193,78,1,1,97,253,
%U A082105 397,481,481,397,253,97,1,1,118,321,526,673,726,673,526,321,118,1
%N A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.
%H A082105 G. C. Greubel, <a href="/A082105/b082105.txt">Antidiagonals n = 0..50, flattened</a>
%F A082105 A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).
%F A082105 A(n, n) = T(2*n, n) = A082106(n) (main diagonal).
%F A082105 T(n, k) = A(n-k, k) (number triangle).
%F A082105 Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).
%F A082105 T(n, n-1) = A028872(n-1), n >= 1.
%F A082105 T(n, n-2) = A082109(n-2), n >= 2.
%F A082105 From _G. C. Greubel_, Dec 22 2022: (Start)
%F A082105 Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).
%F A082105 T(2*n+1, n+1) = A047673(n+1), n >= 0.
%F A082105 T(n, n-k) = T(n, k). (End)
%e A082105 Array, A(n, k), begins as:
%e A082105   1,  1,   1,   1,   1,   1, ... A000012;
%e A082105   1,  6,  13,  22,  33,  46, ... A028872;
%e A082105   1, 13,  33,  61,  97, 141, ... A082109;
%e A082105   1, 22,  61, 118, 193, 286, ... ;
%e A082105   1, 33,  97, 193, 321, 481, ... ;
%e A082105   1, 46, 141, 286, 481, 726, ... ;
%e A082105 Triangle, T(n, k), begins as:
%e A082105   1;
%e A082105   1,  1;
%e A082105   1,  6,   1;
%e A082105   1, 13,  13,   1;
%e A082105   1, 22,  33,  22,   1;
%e A082105   1, 33,  61,  61,  33,   1;
%e A082105   1, 46,  97, 118,  97,  46,   1;
%e A082105   1, 61, 141, 193, 193, 141,  61,  1;
%e A082105   1, 78, 193, 286, 321, 286, 193, 78,  1;
%t A082105 T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1;
%t A082105 Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2022 *)
%o A082105 (Magma) [(k*(n-k))^2 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // _G. C. Greubel_, Dec 22 2022
%o A082105 (SageMath)
%o A082105 def A082105(n,k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1
%o A082105 flatten([[A082105(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 22 2022
%Y A082105 Cf. A016897, A028872, A047673, A082043, A082046, A082106, A082107, A082109.
%K A082105 easy,nonn,tabl
%O A082105 0,5
%A A082105 _Paul Barry_, Apr 03 2003