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Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.
15

%I #438 Sep 27 2024 05:44:21

%S 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,

%T 1,1,37,61,73,73,61,37,1,1,43,73,91,97,91,73,43,1,1,49,85,109,121,121,

%U 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

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

%C From _Kolosov Petro_, Apr 12 2020: (Start)

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

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

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

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

%H Georg Fischer, <a href="/A287326/b287326.txt">Table of n, a(n) for n = 0..495</a> [rows 0..10 and 12..30 from Kolosov Petro]

%H Petro Kolosov, <a href="https://arxiv.org/abs/1603.02468">On the link between Binomial Theorem and Discrete Convolution of Power Function</a>, arXiv:1603.02468 [math.NT], 2016-2020.

%H Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentityInvolvingBTandFaulhaber.pdf">Polynomial identity involving binomial theorem and Faulhaber's formula</a>, 2023.

%H Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.

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

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

%F 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]

%F From _Kolosov Petro_, Jun 05 2019: (Start)

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

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

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

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

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

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

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

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

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

%F 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

%e Triangle begins:

%e ----------------------------------------

%e k= 0 1 2 3 4 5 6 7 8

%e ----------------------------------------

%e n=0: 1;

%e n=1: 1, 1;

%e n=2: 1, 7, 1;

%e n=3: 1, 13, 13, 1;

%e n=4: 1, 19, 25, 19, 1;

%e n=5: 1, 25, 37, 37, 25, 1;

%e n=6: 1, 31, 49, 55, 49, 31, 1;

%e n=7: 1, 37, 61, 73, 73, 61, 37, 1;

%e n=8: 1, 43, 73, 91, 97, 91, 73, 43, 1;

%p T := (n, k) -> 6*k*(n-k) + 1:

%p seq(seq(T(n, k), k=0..n), n=0..11); # _Muniru A Asiru_, Oct 09 2018

%t 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 *)

%o (PARI) t(n, k) = 6*k*(n-k)+1

%o trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))

%o /* Print initial 9 rows of triangle as follows */

%o trianglerows(9) \\ _Felix Fröhlich_, Jan 09 2018

%o (GAP) Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # _Muniru A Asiru_, Oct 09 2018

%o (Magma) /* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Oct 26 2018

%o (SageMath)

%o def A287326(n,k): return 6*k*(n-k) + 1

%o flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 25 2024

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

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

%Y Row sums give A001093.

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

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

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

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

%Y Cf. A294317, A302971, A304042.

%K nonn,tabl,easy

%O 0,5

%A _Kolosov Petro_, Aug 31 2017