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A198063 Triangle read by rows (n >= 0, 0 <= k <= n, m = 3); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j). 8
0, 1, 1, 8, 4, 8, 27, 15, 15, 27, 64, 40, 32, 40, 64, 125, 85, 65, 65, 85, 125, 216, 156, 120, 108, 120, 156, 216, 343, 259, 203, 175, 175, 203, 259, 343, 512, 400, 320, 272, 256, 272, 320, 400, 512, 729, 585, 477, 405, 369, 369, 405, 477, 585, 729 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Read as an infinite symmetric square array, this is the table A(n,k)=(n+k)(n^2+k^2), cf. A321500 for the triangle with k <= n. - M. F. Hasler, Nov 22 2018

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

T(n,k) = 2*k^2*n - 2*k*n^2 + n^3.

T(n,0) = T(n,n) = n^m = n^3 = A000578(n).

T(2*n,n) = (m+1)n^m = 4*n^3 = A033430(n).

T(2*n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^4 - n^4 = A005917(n).

Sum{k=0..n} T(n,k) = (2*n^4 + 3*n^3 + n^2)/3 = A098077(n).

T(n+1,k+1)*C(n,k)^4/(k+1)^3 = A197653(n,k).

EXAMPLE

[0]                   0

[1]                  1, 1

[2]                8, 4, 8

[3]             27, 15, 15, 27

[4]           64, 40, 32, 40, 64

[5]        125, 85, 65, 65, 85, 125

[6]   216, 156, 120, 108, 120, 156, 216

[7] 343, 259, 203, 175, 175, 203, 259, 343

From M. F. Hasler, Nov 22 2018: (Start)

Can also be seen as the square array A(n,k)=(n+k)*(n^2 + k^2) read by antidiagonals:

n | k: 0   1   2   3 ...

--+----------------------

0 |    0   1   8  27 ...

1 |    1   4  15  40 ...

2 |    8  15  32  65 ...

3 |   27  40  65 108 ...

...      ...     ...

(End)

MAPLE

A198063 := (n, k) -> 2*k^2*n-2*k*n^2+n^3:

MATHEMATICA

t[n_, k_] := 2 k^2*n - 2 k*n^2 + n^3; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

PROG

(PARI) A198063(n, k)=2*k^2*n-2*k*n^2+n^3 \\ See also A321500. - M. F. Hasler, Nov 22 2018

(MAGMA) [[2*k^2*n-2*k*n^2+n^3: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 23 2018

(Sage) [[ 2*k^2*n-2*k*n^2+n^3 for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 23 2018

CROSSREFS

Cf. A057427, A003056, A073254, A198064, A198065.

Sequence in context: A200224 A124012 A000803 * A323736 A093208 A155064

Adjacent sequences:  A198060 A198061 A198062 * A198064 A198065 A198066

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Oct 26 2011

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)