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A300785 Triangle read by rows: T(n,k) = 140*k^3*(n-k)^3 - 14*k*(n-k) + 1; n >= 0, 0 <= k <= n. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

From Kolosov Petro, Apr 12 2020: (Start)

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

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

Then T(n, k) = L(3, n, k).

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

LINKS

Muniru A Asiru, Rows n=0..100 of triangle, flattened.

Kolosov Petro, On the link between Binomial Theorem and Discrete Convolution of Power Function, arXiv:1603.02468 [math.NT], 2016-2020.

FORMULA

From Kolosov Petro, Apr 12 2020: (Start)

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

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

EXAMPLE

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;

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 *)

PROG

(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)

(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

(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

(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

CROSSREFS

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

Row sums give A258806.

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

Sequence in context: A025037 A281478 A212927 * A051335 A186995 A145586

Adjacent sequences:  A300782 A300783 A300784 * A300786 A300787 A300788

KEYWORD

nonn,tabl,easy

AUTHOR

Kolosov Petro, Mar 12 2018

STATUS

approved

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Last modified September 30 12:21 EDT 2020. Contains 337439 sequences. (Running on oeis4.)