(* _Peter J. C. Moses, _, Dec 30 2011 *)

_Clark Kimberling (ck6(AT)evansville.edu), _, Dec 31 2011

proposed

approved

editing

proposed

allocated for Clark KimberlingSelf-generating triangle based on symmetric functions.

2, 1, 2, 1, 3, 2, 1, 6, 11, 6, 1, 24, 191, 564, 396, 1, 1176, 435503, 52853928, 1076228496, 1023808896, 1, 2153328000, 1213787658541781999, 58766849935745220643571376, 25431652043775702966453113185344, 29851714119640536870115136698893312

1,1

Let row n+1 be (c0, c1, c2,...,cn). Then

c0*x^n + c1*x^(n-1) +...+ cn=(x+b0)(x+b1)...(x+bm),

where (b0,b1,b2,...,bm) is row n.

row n+1 : f(0,r), f(1,r),...f(n,r), where f(k,r)=(k-th elementary symmetric function), r=(row n).

First five rows:

2

1....2

1....3......2

1....6......11......6

1....24....191....564....396

The factorization property is illustrated by

x^2 + 3x + 2 -> (x+1)(x+3)(x+2) = x^3 + 6x^2 + 11x + 6.

s =.; s[1] = {2};

Prepend[Table[s[z] = Table[SymmetricPolynomial

[k, s[z - 1]], {k, 0, z - 1}], {z, 2, 7}], s[1]]

% // TableForm (* A203301 triangle *)

%% // Flatten (* A203301 sequence *)

(* Peter Moses, Dec 30 2011 *)

Cf. A203300.

allocated

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Dec 31 2011

approved

editing

allocated for Clark Kimberling

allocated

approved