A125166 Triangle, companion to A125165, left border = n^3.

1, 8, 1, 27, 9, 1, 64, 36, 10, 1, 125, 100, 46, 11, 1, 216, 225, 146, 57, 12, 1, 343, 441, 371, 203, 69, 13, 1, 512, 784, 812, 574, 272, 82, 14, 1, 729, 1296, 1596, 1386, 846, 354, 96, 15, 1

Offset

0,2

Comments

Column next to left border = (1, 9, 36, 100, 225...) squares of triangular numbers. A125165 uses analogous operations with n^2 on the left border instead of n^3. Row sums = 1, 9, 37, 111, 283, 657...a sequence analogous to row sums for A125165; i.e. A050488: (1, 5, 15, 37, 83, 177...).

Riordan array ((1+4*x+x^2)/(1-x)^4, x/(1-x)). - Philippe Deléham, Dec 09 2013

Links

Table of n, a(n) for n=0..44.

Formula

Binomial transform of an infinite matrix M with diagonal D, subdiagonal (D-1)..., etc; as follows: (D) = (1,1,1...); (D-1) = (7,7,7...); (D-2) = (12,12,12...); (D-3) = (6,6,6...). Alternatively, given left border n^3: (1, 8, 27, 64...); for k>1, T(n,k) = (n-1,k) + (n-1,k-1).

Example

(5,3) = 146 = (4,3) + (4,2) = 46 + 100.
First few rows of the triangle are:
1;
8, 1;
27, 9, 1;
64, 36, 10, 1;
125, 100, 46, 11, 1;
216, 225, 146, 57, 12, 1;
343, 441, 371, 203, 69, 13, 1;
512, 784, 812, 574, 272, 82, 14, 1;
...

Crossrefs

Cf. A000578, A125165, A050488.

Sequence in context: A002173 A050458 A358877 * A211785 A075151 A075155

Adjacent sequences: A125163 A125164 A125165 * A125167 A125168 A125169

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 22 2006

Extensions

a(27) corrected by Georg Fischer, Feb 18 2020

Status

approved