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

%I A164309 #8 Jun 16 2016 23:27:40
%S A164309 1,5,2,25,10,4,10,125,50,20,50,20,8,20,50,625,250,100,250,100,40,100,
%T A164309 250,100,40,16,40,100,40,100,250
%N A164309 Triangle read by rows, generated from the binomial expansion of (5x + 2).
%C A164309 Row sums = powers of 7: (1, 7, 49, 343,...).
%C A164309 (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.
%F A164309 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;...).
%F A164309 Starting with a power of 5 at left, next term to the right = (5/2)*(current term) if the A164057 term = "1".
%F A164309 If the A164057 term = 0, the next term of A164309 = (2/5)*(current term).
%e A164309 First we construct a 5^n * 2^n array:
%e A164309 .
%e A164309 1...2...4...8...16...
%e A164309 5..10..20..40........
%e A164309 25.50................
%e A164309 125..................
%e A164309 .
%e A164309 Extract diagonal terms: (1; 5,2; 25,10,4;...) then use the multiplication rules given in the formulas section.
%e A164309 .
%e A164309 First few rows of triangle A164057:
%e A164309 1;
%e A164309 1, 0;
%e A164309 1, 0, 0, 1;
%e A164309 1, 0, 0, 1, 0, 0, 1, 1;
%e A164309 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1;
%e A164309 ...
%e A164309 Using the rules, we obtain:
%e A164309 .
%e A164309 1;
%e A164309 5, 2;
%e A164309 25, 10, 4, 10;
%e A164309 125, 50, 20, 50, 20, 8, 20, 50;
%e A164309 625, 250, 100, 250, 100, 40, 100, 250, 100, 40, 16, 40, 100, 40, 100, 250;
%e A164309 ...
%e A164309 Example: place row 3 of A164056 on top of construction of row 3, A164309:
%e A164309 .
%e A164309 (1,...0,...0,...1,...0,...0,...1,...1):
%e A164309 (125,50,..20,..50,..20,...8,..20,..50)
%e A164309 .
%e A164309 "50) = (2/5)*125, while 50 = (5/2)*20; etc.
%Y A164309 Cf. A164057, A164308.
%K A164309 nonn,tabl
%O A164309 0,2
%A A164309 _Gary W. Adamson_, Aug 12 2009