A300785 - OEIS

%I #109 Sep 14 2024 12:31:19

%S 1,1,1,1,127,1,1,1093,1093,1,1,3739,8905,3739,1,1,8905,30157,30157,

%T 8905,1,1,17431,71569,101935,71569,17431,1,1,30157,139861,241753,

%U 241753,139861,30157,1,1,47923,241753,472291,573217,472291,241753,47923,1,1,71569,383965,816229,1119721,1119721,816229,383965,71569,1

%N Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*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(3, n, k).

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

%H Muniru A Asiru, <a href="/A300785/b300785.txt">Rows n=0..100 of triangle, flattened</a>.

%H Kolosov Petro, <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], 2016-2022.

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

%F T(n, k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1.

%F T(n, k) = 140*A094053(n, k)^3 + 0*A094053(n, k)^2 - 14*A094053(n, k)^1 + 1.

%F T(n+3, k) = 4*T(n+2, k) - 6*T(n+1, k) + 4*T(n, k) - T(n-1, k), for n >= k.

%F Sum_{k=1..n} T(n, k) = A001015(n).

%F Sum_{k=0..n} T(n, k) = A258806(n).

%F Sum_{k=0..n-1} T(n, k) = A001015(n).

%F Sum_{k=1..n-1} T(n, k) = A258808(n).

%F Sum_{k=1..n-1} T(n, k) = -A024005(n).

%F Sum_{k=1..r} T(n, k) = -A316387(3, r, 0)*n^0 + A316387(3, r ,1)*n^1 - A316387(3, r, 2)*n^2 + A316387(3, r, 3)*n^3. (End)

%F G.f.: ((1 + 127*x^6*y^3 - 3*x*(1 + y) + 585*x^5*y^2*(1 + y) + 129*x^4*y*(1 + 17*y + y^2) + 3*x^2*(1 + 45*y + y^2) - x^3*(1 - 579*y - 579*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - _Stefano Spezia_, Sep 14 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,   127,      1;

%e n=3: 1,  1093,   1093,      1;

%e n=4: 1,  3739,   8905,   3739,      1;

%e n=5: 1,  8905,  30157,  30157,   8905,      1;

%e n=6: 1, 17431,  71569, 101935,  71569,  17431,      1;

%e n=7: 1, 30157, 139861, 241753, 241753, 139861,  30157,     1;

%e n=8: 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923,    1;

%p T:=(n,k)->140*k^3*(n-k)^3-14*k*(n-k)+1: seq(seq(T(n,k),k=0..n),n=0..9); # _Muniru A Asiru_, Dec 14 2018

%t T[n_, k_] := 140*k^3*(n - k)^3 - 14*k*(n - k) + 1; Column[

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* From _Kolosov Petro_, Apr 12 2020 *)

%o (PARI) t(n, k) = 140*k^3*(n-k)^3-14*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 */ trianglerows(9)

%o (Magma) /* As triangle */ [[140*k^3*(n-k)^3-14*k*(n-k)+1: k in [0..n]]: n in [0..10]]; // _Bruno Berselli_, Mar 21 2018

%o (Sage) [[140*k^3*(n-k)^3 - 14*k*(n-k)+1 for k in range(n+1)] for n in range(12)] # _G. C. Greubel_, Dec 14 2018

%o (GAP) T:=Flat(List([0..9], n->List([0..n], k->140*k^3*(n-k)^3 - 14*k*(n-k)+1))); # _G. C. Greubel_, Dec 14 2018

%Y Various cases of L(m, n, k): A287326 (m=1), A300656 (m=2), This sequence (m=3). See comments for L(m, n, k).

%Y Row sums give A258806.

%Y Cf. A000584, A287326, A007318, A077028, A294317, A068236, A300656, A302971, A304042, A001015, A094053, A258808, A024005, A316387.

%K nonn,tabl,easy

%O 0,5

%A _Kolosov Petro_, Mar 12 2018