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.

1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500

Offset

1,2

Comments

T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).

Links

Table of n, a(n) for n=1..55.

Formula

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).

T(n,2) = A123865(n-1) for n > 1.

Example

n\k|    0    1    2    3    4    5
---+------------------------------
1  |    1
2  |   16    4
3  |   81   36   15
4  |  256  144   80   32
5  |  625  400  255  140   65
6  | 1296  900  624  396  240  108
      ...
For n = 3 the matrix M is
  [ 1,  4, 15]
  [ 4, 15, 32]
  [15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.

Mathematica

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

Crossrefs

Cf. A317614 (diagonal), A374668.

Cf. A333119, A330613.

Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).

Sequence in context: A331227 A040247 A245826 * A331222 A089083 A363191

Adjacent sequences: A374705 A374706 A374707 * A374709 A374710 A374711

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Jul 17 2024

Status

approved