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A276189
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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.
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2
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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, 970, 1208, 1302, 1180, 770, 10, 126, 400, 770, 1164, 1510, 1736, 1770, 1540, 974, 11, 140, 450, 880, 1358, 1812, 2170, 2360, 2310, 1948, 1202
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
...
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MATHEMATICA
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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 *)
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PROG
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(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]];
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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