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, 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

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

Cf. A000583, A005914, A276158.

Sequence in context: A298800 A138907 A336837 * 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

a(40) ff. corrected by Georg Fischer, Nov 08 2021

Status

approved