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A276189 Triangle read by rows: T(n,k) = 2*(6*k^2 + 1)*(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1. 1
1, 2, 14, 3, 28, 50, 4, 42, 100, 110, 5, 56, 150, 220, 194, 6, 70, 200, 330, 388, 302, 7, 84, 250, 440, 582, 604, 434, 8, 98, 300, 550, 776, 906, 868, 590, 9, 112, 350, 660, 776, 1208, 1302, 920, 770, 10, 126, 400, 770, 970, 1510, 1736, 1250, 1540, 974 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The row sums of the triangle provide the positive terms of A000583.

Similar triangles can be generated by the formula P(n,k,m) = (Q(k+1,m)-Q(k,m))*(n+1-k), where Q(i,r) = i^r-(i-1)^r, 0 < k <= n, and P(n,0,m) = n+1. T(n,k) is the case m=4, that is T(n,k) = P(n,k,4).

LINKS

Table of n, a(n) for n=0..54.

FORMULA

T(n,n-h) = (h+1)*A005914(n-h) for 0 <= h <= n. Therefore, the main diagonal of the triangle is A005914.

Sum_{k=0..n} T(n,k) = T(n,0)^4 = A000583(n+1).

EXAMPLE

Triangle starts:

----------------------------------------------

n \ k |  0   1    2    3    4    5    6    7

----------------------------------------------

0     |  1;

1     |  2, 14;

2     |  3, 28,  50;

3     |  4, 42, 100, 110;

4     |  5, 56, 150, 220, 194;

5     |  6, 70, 200, 330, 388, 302;

6     |  7, 84, 250, 440, 582, 604, 434;

7     |  8, 98, 300, 550, 776, 906, 868, 590;

...

MATHEMATICA

Table[If[k == 0, n + 1, 2 (6 k^2 + 1) (n + 1 - k)], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 29 2016 *)

PROG

(MAGMA) [IsZero(k) select n+1 else 2*(6*k^2+1)*(n+1-k): k in [0..n], n in [0..10]];

(MAGMA) /* As triangle (see the second comment): */ m:=4; Q:=func<i, r | i^r-(i-1)^r>; P:=func<n, k, m | IsZero(k) select n+1 else (Q(k+1, m)-Q(k, m))*(n+1-k)>; [[P(n, k, m): k in [0..n]]: n in [0..10]];

CROSSREFS

Cf. A000583, A005914, A276158.

Sequence in context: A227628 A298800 A138907 * A281944 A306724 A316909

Adjacent sequences:  A276186 A276187 A276188 * A276190 A276191 A276192

KEYWORD

nonn,tabl

AUTHOR

Stefano Maruelli, Aug 24 2016

EXTENSIONS

Corrected, rewritten and extended by Bruno Berselli, Aug 31 2016

STATUS

approved

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Last modified February 29 08:26 EST 2020. Contains 332355 sequences. (Running on oeis4.)