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%I #6 Dec 04 2016 19:46:26
%S 1,1,1,1,2,1,1,4,5,2,1,12,49,78,40,1,180,11085,270610,2094264,1834560,
%T 1,4210700,4952544856489,1094968722994345590,11723079808649412379800,
%U 2086231309557403469400000,2074509324712524510720000
%N Self-generating triangle based on symmetric functions.
%C Let row n+1 be (c0, c1, c2,...,cn). Then
%C c0*x^n + c1*x^(n-1) +...+ cn=(x+b0)(x+b1)...(x+bm),
%C where (b0,b1,b2,...,bm) is row n.
%F row n+1: f(0,r), f(1,r),...f(n,r), where f(k,r)=(k-th elementary symmetric function), r=(row n).
%e First five rows:
%e 1
%e 1...1
%e 1...2....1
%e 1...4....5....2
%e 1...12...49...78...40
%e The factorization property is illustrated by
%e x^2 + 2x + 1 -> (x+1)(x+2)(x+1) = x^3 + 4x^2 + 5x + 2.
%t s =.; s[1] = {1};
%t Prepend[Table[s[z] = Table[SymmetricPolynomial
%t [k, s[z - 1]], {k, 0, z - 1}], {z, 2, 7}], s[1]]
%t % // TableForm (* A203300 triangle *)
%t %% // Flatten (* A203300 sequence *)
%t (* _Peter J. C. Moses_, Dec 30 2011 *)
%Y Cf. A203301.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Dec 31 2011
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