A287326 Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1

Offset: 0

Author

Kolosov Petro, Aug 31 2017

Keywords

• nonn
• tabl
• easy

Comments

From Kolosov Petro, Apr 12 2020: (Start)

Let A(m, r) = A302971(m, r) / A304042(m, r).

Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.

Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.

T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

Examples

			Triangle begins:
  ----------------------------------------
  k=    0   1   2   3   4   5   6   7   8
  ----------------------------------------
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  7,  1;
  n=3:  1, 13, 13,  1;
  n=4:  1, 19, 25, 19,  1;
  n=5:  1, 25, 37, 37, 25,  1;
  n=6:  1, 31, 49, 55, 49, 31,  1;
  n=7:  1, 37, 61, 73, 73, 61, 37,  1;
  n=8:  1, 43, 73, 91, 97, 91, 73, 43,  1;
		

Links

• Georg Fischer, Table of n, a(n) for n = 0..495 [rows 0..10 and 12..30 from Kolosov Petro]
• Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
• Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
• Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Crossrefs

Columns k=0..6 give A000012, A016921, A017533, A161705, A103214, A128470, A158065.

Column sums k=0..4 give A000027, A000567, A051866, A051872, A255185.

Row sums give A001093.

Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).

Differences of cubes n^3 are T(A000124(n), 1).

Cf. A000124, A000578, A003154, A003215, A007318, A008458, A016969.

Cf. A038593, A055012, A068601, A077028, A094053, A166873, A227776.

Cf. A294317, A302971, A304042.

Programs

GAP

Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018

Magma

/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018

Maple

T := (n, k) -> 6*k*(n-k) + 1:
seq(seq(T(n, k), k=0..n), n=0..11); # Muniru A Asiru, Oct 09 2018

Mathematica

T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)

PARI

t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
trianglerows(9) \\ Felix Fröhlich, Jan 09 2018

SageMath

def A287326(n,k): return 6*k*(n-k) + 1
flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 25 2024

Formula

T(n, k) = 6*k*(n-k) + 1.

G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.

G.f.: (1 - x - x*y + 7*x^2*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Oct 09 2018 [Adapted by Stefano Spezia, Sep 25 2024]

From Kolosov Petro, Jun 05 2019: (Start)

T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.

T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.

T(n, k) = 6*A077028(n, k) - 5.

T(2n, n) = A227776(n).

T(2n+1, n) = A003154(n+1).

T(2n+3, n) = A166873(n+1).

Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).

Sum_{k=1..n-1} T(n, k) = A068601(n).

(n+1)^3 - n^3 = T(A000124(n), 1). (End)

Sum_{k=0..n} (-1)^k*T(n, k) = (-1/2)*(1 + (-1)^n)*A016969(floor(n/2) - 1). - G. C. Greubel, Sep 25 2024