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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. 2
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 (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
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
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
Sequence in context: A298800 A138907 A336837 * A281944 A306724 A316909
KEYWORD
nonn,tabl
AUTHOR
Stefano Maruelli, Aug 24 2016
EXTENSIONS
Corrected, rewritten and extended by Bruno Berselli, Aug 31 2016
a(40) ff. corrected by Georg Fischer, Nov 08 2021
STATUS
approved

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Last modified April 19 11:14 EDT 2024. Contains 371791 sequences. (Running on oeis4.)