A325516 Triangle read by rows: T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2), with 0 <= k < n.
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, 35, 14, 7, 256, 200, 144, 104, 64, 40, 16, 8, 369, 288, 225, 162, 117, 72, 45, 18, 9, 500, 410, 320, 250, 180, 130, 80, 50, 20, 10, 671, 550, 451, 352, 275, 198, 143, 88, 55, 22, 11
Offset: 1
Examples
The triangle T(n, k) begins: ---+------------------------------ n\k| 0 1 2 3 4 ---+------------------------------ 1 | 1 2 | 4 2 3 | 15 6 3 4 | 32 20 8 4 5 | 65 40 25 10 5 ... 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) = 2 + 4 = 6, and T(3, 2) = 3.
Programs
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GAP
Flat(List([1..11], n->List([0..n-1], k->(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2))));
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Magma
[[(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2): k in [0..n-1]]: n in [1..11]];
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Maple
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);
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Mathematica
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}]] -
PARI
T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^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 + 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)).
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)).
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.
1st column: T(n, 1) = A317614(n).
Diagonal: T(n, n-1) = n.
Comments