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A325655 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.

%I A325655 #12 Sep 08 2022 08:46:24
%S A325655 1,4,4,15,14,7,32,32,24,16,65,64,53,42,21,108,108,96,84,60,36,175,174,
%T A325655 159,144,115,86,43,256,256,240,224,192,160,112,64,369,368,349,330,293,
%U A325655 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 A325655 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 A325655 T(n, k) is the k-subdiagonal sum of the matrix M(n) whose permanent is A322277(n).
%F A325655 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 A325655 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 A325655 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 A325655 Diagonal: T(n, n-1) = A325657(n).
%F A325655 1st column: T(n, 0) = A317614(n).
%e A325655 The triangle T(n, k) begins:
%e A325655 ---+-----------------------------
%e A325655 n\k|    0     1     2     3     4
%e A325655 ---+-----------------------------
%e A325655 1  |    1
%e A325655 2  |    4     4
%e A325655 3  |   15    14     7
%e A325655 4  |   32    32    24    16
%e A325655 5  |   65    64    53    42    21
%e A325655 ...
%e A325655 For n = 3 the matrix M(3) is
%e A325655     1, 2, 3
%e A325655     6, 5, 4
%e A325655     7, 8, 9
%e A325655 and therefore T(3, 0) = 1 + 5 + 9 = 15, T(3, 1) = 6 + 8 = 14, and T(3, 2) = 7.
%p A325655 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 A325655 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 A325655 (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 A325655 (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 A325655 (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 A325655 tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print);
%o A325655 tabl(11) \\ yields sequence in triangular form
%Y A325655 Cf. A317614, A322277, A323723 (k = 1), A325656 (row sums), A325657 (diagonal).
%K A325655 nonn,tabl,easy
%O A325655 1,2
%A A325655 _Stefano Spezia_, May 13 2019