%I #12 Sep 08 2022 08:46:24

%S 1,4,4,15,14,7,32,32,24,16,65,64,53,42,21,108,108,96,84,60,36,175,174,

%T 159,144,115,86,43,256,256,240,224,192,160,112,64,369,368,349,330,293,

%U 256,201,146,73,500,500,480,460,420,380,320,260,180,100,671,670,647,624,579,534,467,400,311,222,111

%N Triangle read by rows: T(n, k) = (1/4)*(2*(-1 + (-1)^n)*k - 2*k^2*n + n*(2 - (-1)^k - (-1)^n + 2*n^2)), with 0 <= k < n.

%C T(n, k) is the k-subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

%F O.g.f.: x*(- 1 + 2*y + 3*y^2 - 2*y^3 + 2*x*(- 1 + y^2) + x^4*(- 1 + 3*y^2) + x^2*(- 6 + 6*y + 2*y^2 - 6*y^3) + x^3*(- 2 + 4*y + 2*y^2 - 4*y^3))/((- 1 + x)^4*(1 + x)^2*(- 1 + y)^3*(1 + y)).

%F E.g.f.: (1/4)*exp(- x - y)*(- exp(2*x)*x + exp(2*y)*(x + 2*y) + 2*exp(2*(x + y))*(3*x^2 + x^3 - y - x*(- 2 + y + y^2))).

%F T(n, k) = (1/2)*n*(n^2 - k^2) if n and k are both even; T(n, k) = (1/2)*n*(n^2 - k^2 + 1) if n is even and k is odd; T(n, k) = (1/2)*(n*(n^2 - k^2 + 1) - 2*k) if n is odd and k is even; T(n, k) = (1/2)*(n*(n^2 - k^2 + 2) - 2*k) if n and k are both odd.

%F Diagonal: T(n, n-1) = A325657(n).

%F 1st column: T(n, 0) = A317614(n).

%e The triangle T(n, k) begins:

%e ---+-----------------------------

%e n\k| 0 1 2 3 4

%e ---+-----------------------------

%e 1 | 1

%e 2 | 4 4

%e 3 | 15 14 7

%e 4 | 32 32 24 16

%e 5 | 65 64 53 42 21

%e ...

%e For n = 3 the matrix M(3) is

%e 1, 2, 3

%e 6, 5, 4

%e 7, 8, 9

%e and therefore T(3, 0) = 1 + 5 + 9 = 15, T(3, 1) = 6 + 8 = 14, and T(3, 2) = 7.

%p a:=(n, k)->(1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^2)): seq(seq(a(n, k), k=0..n-1), n=1..11);

%t T[n_, k_]:=(1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^2)); Flatten[Table[T[n,k],{n,1,11},{k,0,n-1}]]

%o (GAP) Flat(List([1..11], n->List([0..n-1], k->(1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^2)))));

%o (Magma) [[(1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^2)): k in [0..n-1]]: n in [1..11]];

%o (PARI) T(n, k) = (1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^2));

%o tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print);

%o tabl(11) \\ yields sequence in triangular form

%Y Cf. A317614, A322277, A323723 (k = 1), A325656 (row sums), A325657 (diagonal).

%K nonn,tabl,easy

%O 1,2

%A _Stefano Spezia_, May 13 2019