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A325516 Triangle read by rows: T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2), with 0 <= k < n.

%I A325516 #22 Sep 08 2022 08:46:24
%S A325516 1,4,2,15,6,3,32,20,8,4,65,40,25,10,5,108,78,48,30,12,6,175,126,91,56,
%T A325516 35,14,7,256,200,144,104,64,40,16,8,369,288,225,162,117,72,45,18,9,
%U A325516 500,410,320,250,180,130,80,50,20,10,671,550,451,352,275,198,143,88,55,22,11
%N A325516 Triangle read by rows: T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2), with 0 <= k < n.
%C A325516 T(n, k) is the k-superdiagonal sum of the matrix M(n) whose permanent is A322277(n).
%F A325516 O.g.f.: x*(1 - 2*y + y^2 + 2*y^3 + x^4*(1 + y^2) + 2*x^3*(1 + y - 3*y^2 + y^3) + 2*x^2*(3 - 5*y - y^2 + y^3) + x*(2 - 2*y - 6*y^2 + 6*y^3))/((1 - x)^4*(1 + x)^2*(1 - y)^3*(1 + y)).
%F A325516 E.g.f.: (1/4)*exp(-x - y)*(x + exp(2*(x + y))*x*(3 + 2*x^2 + x*(6 - 4*y) - 2*y + 2*y^2)).
%F A325516 T(n, k) = n*(n - k)^2/2 if n and k are both even or odd; T(n, k) = n*(n - k)^2/2 + n/2 otherwise.
%F A325516 1st column: T(n, 1) = A317614(n).
%F A325516 Diagonal: T(n, n-1) = n.
%e A325516 The triangle T(n, k) begins:
%e A325516 ---+------------------------------
%e A325516 n\k|    0     1     2     3     4
%e A325516 ---+------------------------------
%e A325516 1  |    1
%e A325516 2  |    4     2
%e A325516 3  |   15     6     3
%e A325516 4  |   32    20     8     4
%e A325516 5  |   65    40    25    10     5
%e A325516 ...
%e A325516 For n = 3 the matrix M(3) is
%e A325516 1, 2, 3
%e A325516 6, 5, 4
%e A325516 7, 8, 9
%e A325516 and therefore T(3, 0) = 1 + 5 + 9 = 15, T(3, 1) = 2 + 4 = 6, and T(3, 2) = 3.
%p A325516 a:=(n, k)->(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2): seq(seq(a(n, k), k=0..n-1), n=1..11);
%t A325516 T[n_,k_] := (1/4) n (1 - (-1)^(n - k) + 2 (n - k)^2); Flatten[Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}]]
%o A325516 (GAP) Flat(List([1..11], n->List([0..n-1], k->(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2))));
%o A325516 (Magma) [[(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2): k in [0..n-1]]: n in [1..11]];
%o A325516 (PARI) T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2);
%o A325516 tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print);
%o A325516 tabl(11) \\ yields sequence in triangular form
%Y A325516 Cf. A000027, A317614, A322277, A325517 (row sums).
%K A325516 nonn,tabl,easy
%O A325516 1,2
%A A325516 _Stefano Spezia_, May 07 2019