A164309 Triangle read by rows, generated from the binomial expansion of (5x + 2).

1, 5, 2, 25, 10, 4, 10, 125, 50, 20, 50, 20, 8, 20, 50, 625, 250, 100, 250, 100, 40, 100, 250, 100, 40, 16, 40, 100, 40, 100, 250

Offset: 0

Author

Gary W. Adamson, Aug 12 2009

Keywords

• nonn
• tabl

Comments

Row sums = powers of 7: (1, 7, 49, 343,...).

(5x + 2)^3 = 125x^3 + 150x^2 +60x + 8; with reference to row 3, since there is one 125, three 50's = 150, three 20's = 60, and one 8 = total 343 = 7^3.

Examples

			First we construct a 5^n * 2^n array:
.
1...2...4...8...16...
5..10..20..40........
25.50................
125..................
.
Extract diagonal terms: (1; 5,2; 25,10,4;...) then use the multiplication rules given in the formulas section.
.
First few rows of triangle A164057:
1;
1, 0;
1, 0, 0, 1;
1, 0, 0, 1, 0, 0, 1, 1;
1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1;
...
Using the rules, we obtain:
.
1;
5, 2;
25, 10, 4, 10;
125, 50, 20, 50, 20, 8, 20, 50;
625, 250, 100, 250, 100, 40, 100, 250, 100, 40, 16, 40, 100, 40, 100, 250;
...
Example: place row 3 of A164056 on top of construction of row 3, A164309:
.
(1,...0,...0,...1,...0,...0,...1,...1):
(125,50,..20,..50,..20,...8,..20,..50)
.
"50) = (2/5)*125, while 50 = (5/2)*20; etc.
		

Crossrefs

Cf. A164057, A164308.

Formula

Given terms in a 2^n * 5^n multiplication table (diagonals of the array); map the terms in a triangle with 2^n terms per row using the template of A164057: (1;, 1,0; 1,0,0,1; 1,0,0,1,0,0,1,1;...).

Starting with a power of 5 at left, next term to the right = (5/2)*(current term) if the A164057 term = "1".

If the A164057 term = 0, the next term of A164309 = (2/5)*(current term).