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A374708 Triangle T read by rows: T(n,k) = (n - k)*n*(4*n^2 - 4*n*k + 2*k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

%I A374708 #11 Jul 19 2024 14:26:11
%S A374708 1,16,4,81,36,15,256,144,80,32,625,400,255,140,65,1296,900,624,396,
%T A374708 240,108,2401,1764,1295,896,609,364,175,4096,3136,2400,1760,1280,864,
%U A374708 544,256,6561,5184,4095,3132,2385,1728,1215,756,369,10000,8100,6560,5180,4080,3100,2320,1620,1040,500
%N A374708 Triangle T read by rows: T(n,k) = (n - k)*n*(4*n^2 - 4*n*k + 2*k^2 - 1 + (-1)^k)/4, with 0 <= k < n.
%C A374708 T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).
%F A374708 O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).
%F A374708 T(n,2) = A123865(n-1) for n > 1.
%e A374708 n\k|    0    1    2    3    4    5
%e A374708 ---+------------------------------
%e A374708 1  |    1
%e A374708 2  |   16    4
%e A374708 3  |   81   36   15
%e A374708 4  |  256  144   80   32
%e A374708 5  |  625  400  255  140   65
%e A374708 6  | 1296  900  624  396  240  108
%e A374708       ...
%e A374708 For n = 3 the matrix M is
%e A374708   [ 1,  4, 15]
%e A374708   [ 4, 15, 32]
%e A374708   [15, 32, 65]
%e A374708 and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.
%t A374708 T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten
%Y A374708 Cf. A317614 (diagonal), A374668.
%Y A374708 Cf. A333119, A330613.
%Y A374708 Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).
%K A374708 nonn,easy,tabl
%O A374708 1,2
%A A374708 _Stefano Spezia_, Jul 17 2024