A082105 - OEIS

%I #14 Dec 24 2022 03:52:51

%S 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 1,1,61,141,193,193,141,61,1,1,78,193,286,321,286,193,78,1,1,97,253,

%U 397,481,481,397,253,97,1,1,118,321,526,673,726,673,526,321,118,1

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

%H G. C. Greubel, <a href="/A082105/b082105.txt">Antidiagonals n = 0..50, flattened</a>

%F A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).

%F A(n, n) = T(2*n, n) = A082106(n) (main diagonal).

%F T(n, k) = A(n-k, k) (number triangle).

%F Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).

%F T(n, n-1) = A028872(n-1), n >= 1.

%F T(n, n-2) = A082109(n-2), n >= 2.

%F From _G. C. Greubel_, Dec 22 2022: (Start)

%F Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).

%F T(2*n+1, n+1) = A047673(n+1), n >= 0.

%F T(n, n-k) = T(n, k). (End)

%e Array, A(n, k), begins as:

%e   1,  1,   1,   1,   1,   1, ... A000012;

%e   1,  6,  13,  22,  33,  46, ... A028872;

%e   1, 13,  33,  61,  97, 141, ... A082109;

%e   1, 22,  61, 118, 193, 286, ... ;

%e   1, 33,  97, 193, 321, 481, ... ;

%e   1, 46, 141, 286, 481, 726, ... ;

%e Triangle, T(n, k), begins as:

%e   1;

%e   1,  1;

%e   1,  6,   1;

%e   1, 13,  13,   1;

%e   1, 22,  33,  22,   1;

%e   1, 33,  61,  61,  33,   1;

%e   1, 46,  97, 118,  97,  46,   1;

%e   1, 61, 141, 193, 193, 141,  61,  1;

%e   1, 78, 193, 286, 321, 286, 193, 78,  1;

%t T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1;

%t Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2022 *)

%o (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 (SageMath)

%o def A082105(n,k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1

%o flatten([[A082105(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 22 2022

%Y Cf. A016897, A028872, A047673, A082043, A082046, A082106, A082107, A082109.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Apr 03 2003