A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1; n >= 0, 0 <= k <= n.
1, 1, 1, 1, 127, 1, 1, 1093, 1093, 1, 1, 3739, 8905, 3739, 1, 1, 8905, 30157, 30157, 8905, 1, 1, 17431, 71569, 101935, 71569, 17431, 1, 1, 30157, 139861, 241753, 241753, 139861, 30157, 1, 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1, 1, 71569, 383965, 816229, 1119721, 1119721, 816229, 383965, 71569, 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, 127, 1; n=3: 1, 1093, 1093, 1; n=4: 1, 3739, 8905, 3739, 1; n=5: 1, 8905, 30157, 30157, 8905, 1; n=6: 1, 17431, 71569, 101935, 71569, 17431, 1; n=7: 1, 30157, 139861, 241753, 241753, 139861, 30157, 1; n=8: 1, 47923, 241753, 472291, 573217, 472291, 241753, 47923, 1;
Links
- Muniru A Asiru, Rows n=0..100 of triangle, flattened.
- 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
-
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
-
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
-
Maple
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
-
Mathematica
T[n_, k_] := 140*k^3*(n - k)^3 - 14*k*(n - k) + 1; Column[ Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* From Kolosov Petro, Apr 12 2020 *) -
PARI
t(n, k) = 140*k^3*(n-k)^3-14*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) -
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
Formula
From Kolosov Petro, Apr 12 2020: (Start)
T(n, k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1.
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.
Sum_{k=1..n} T(n, k) = A001015(n).
Sum_{k=0..n} T(n, k) = A258806(n).
Sum_{k=0..n-1} T(n, k) = A001015(n).
Sum_{k=1..n-1} T(n, k) = A258808(n).
Sum_{k=1..n-1} T(n, k) = -A024005(n).
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)
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
Comments