OFFSET
0,5
COMMENTS
From Kolosov Petro, Apr 12 2020: (Start)
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)
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 of Power Function, arXiv:1603.02468 [math.NT], 2016-2020.
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.
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
EXAMPLE
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;
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Kolosov Petro, Aug 31 2017
STATUS
approved