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

1,2

Comments

T(n, k) is the k-subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

Links

Table of n, a(n) for n=1..66.

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).

Example

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.

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);

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}]]

Prog

(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)))));
(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]];
(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

Crossrefs

Cf. A317614, A322277, A323723 (k = 1), A325656 (row sums), A325657 (diagonal).

Sequence in context: A270844 A287286 A271546 * A117187 A138767 A048282

Adjacent sequences: A325652 A325653 A325654 * A325656 A325657 A325658

Keyword

nonn,tabl,easy

Author

Stefano Spezia, May 13 2019

Status

approved