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.
1, 4, 4, 15, 14, 7, 32, 32, 24, 16, 65, 64, 53, 42, 21, 108, 108, 96, 84, 60, 36, 175, 174, 159, 144, 115, 86, 43, 256, 256, 240, 224, 192, 160, 112, 64, 369, 368, 349, 330, 293, 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
Offset: 1
Examples
The triangle T(n, k) begins:
---+-----------------------------
n\k| 0 1 2 3 4
---+-----------------------------
1 | 1
2 | 4 4
3 | 15 14 7
4 | 32 32 24 16
5 | 65 64 53 42 21
...
For n = 3 the matrix M(3) is
1, 2, 3
6, 5, 4
7, 8, 9
and therefore T(3, 0) = 1 + 5 + 9 = 15, T(3, 1) = 6 + 8 = 14, and T(3, 2) = 7.
Programs
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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)))));
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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]];
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Maple
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);
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Mathematica
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}]] -
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)); tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); tabl(11) \\ yields sequence in triangular form
Formula
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)).
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))).
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.
Diagonal: T(n, n-1) = A325657(n).
1st column: T(n, 0) = A317614(n).
Comments