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A160770

Triangle in which row n is generated from (1,3,6,10,...n) dot (n,n-1,...1) with subtractive carryovers.

1, 3, 0, 6, 3, 3, 10, 8, 10, 0, 15, 15, 21, 9, 6, 21, 24, 36, 24, 21, 0, 28, 35, 55, 45, 45, 18, 10, 36, 48, 78, 72, 78, 48, 36, 0, 45, 63, 105, 105, 120, 90, 78, 30, 15

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OFFSET

0,2

COMMENTS

Row sums = A005995: (1, 3, 12, 28, 66, 126, 236,...); also generated from:

(1/2)*((1, 6, 21, 56, 126,...)+(1, 0, 3, 0, 6, 0, 10,...)); where (1, 6, 21,...) = bin(n,5).

A177878 = the analogous sequence using vector (1,2,3,...).

LINKS

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

FORMULA

The subtractive carryover dot product of two vectors (a(1),a(2),...a(n)) dot (b(1),b(2),..b(n)) = (c(1),..,c(n)) is defined by c(1)=a(1)*b(1) and c(i) = a(i)*b(i)-c(i-1), i>1; where the vector = the triangular series.

EXAMPLE

First few rows of the triangle:

3, 0;

6, 3, 3;

10, 8, 10, 0;

15, 15, 21, 9, 6;

21, 24, 36, 24, 21, 0;

28, 35, 55, 45, 45, 18, 10;

36, 48, 78, 72, 78, 48, 36, 0

45, 63, 105, 105, 120, 90, 78, 30, 15;

...

Example: row 2 = (6, 3, 3) = (1, 3, 6) dot (6, 3, 1) with subtractive carryovers = ((1*6=6), (3*3-6=3), (6*1-3=3)) = (6, 3, 3).