A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

0, 0, 1, 0, 4, 5, 0, 7, 17, 12, 0, 10, 29, 39, 22, 0, 13, 41, 66, 70, 35, 0, 16, 53, 93, 118, 110, 51, 0, 19, 65, 120, 166, 185, 159, 70, 0, 22, 77, 147, 214, 260, 267, 217, 92, 0, 25, 89, 174, 262, 335, 375, 364, 284, 117, 0, 28, 101, 201, 310, 410, 483, 511, 476, 360, 145

Offset

0,5

Comments

The main diagonal is A024394.

The antidiagonals sums are A000537.

Links

Table of n, a(n) for n=0..65.

Formula

Take successively sequences n*(3*n-1)/2, n*(9*n-1)/2, n*(15*n-1)/2, n*(21*n-1)/2, ... listed in the EXAMPLE section.

From Stefano Spezia, Feb 21 2024: (Start)

G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).

E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)

Example

The rows are:
  0  1  5  12  22  35  51  70 ... = A000326
  0  4 17  39  70 110 159 217 ... = A022266
  0  7 29  66 118 185 267 364 ... = A022272
  0 10 41  93 166 260 375 511 ... = A022278
  0 13 53 120 214 335 483 658 ... = A022284
  ... .
Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).
Difference between two consecutive rows are: A033428.
This square array read by antidiagonals leads to the triangle
  0
  0  1
  0  4  5
  0  7 17 12
  0 10 29 39  22
  0 13 41 66  70  35
  0 16 53 93 118 110 51
  ... .

Maple

T:= (n, k)-> k*(k*(3+6*n)-1)/2:
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2023

Mathematica

T[n_, k_] := ((6*n + 3)*k - 1)*k/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 27 2023 *)

Prog

(PARI) T(n, k) = k*((3+6*n)*k-1)/2; \\ Michel Marcus, Feb 27 2023

Crossrefs

Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.

Cf. A033428.

Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.

Sequence in context: A318740 A240160 A249860 * A320162 A354068 A200013

Adjacent sequences: A360959 A360960 A360961 * A360963 A360964 A360965

Keyword

nonn,tabl

Author

Paul Curtz, Feb 27 2023

Status

approved