OFFSET
1,2
COMMENTS
T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).
LINKS
Stefano Spezia, First 150 rows of the triangle, flattened
FORMULA
O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]
EXAMPLE
n\k| 0 1 2 3 4 5
---+------------------------
1 | 1
2 | 5 2
3 | 13 8 3
4 | 25 18 11 4
5 | 41 32 23 14 5
6 | 61 50 39 28 17 6
...
For n = 3 the matrix M is
1, 2, 3
2, 4, 6
3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.
MATHEMATICA
Flatten[Table[1+k-2n-2k*n+2n^2, {n, 1, 11}, {k, 0, n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]
CROSSREFS
KEYWORD
AUTHOR
Stefano Spezia, Dec 20 2019
STATUS
approved