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
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
Programs
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GAP
Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018
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Magma
/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018
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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
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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
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