A203300 Self-generating triangle based on symmetric functions.

1, 1, 1, 1, 2, 1, 1, 4, 5, 2, 1, 12, 49, 78, 40, 1, 180, 11085, 270610, 2094264, 1834560, 1, 4210700, 4952544856489, 1094968722994345590, 11723079808649412379800, 2086231309557403469400000, 2074509324712524510720000

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..28.

Formula

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

Example

First five rows:
1
1...1
1...2....1
1...4....5....2
1...12...49...78...40
The factorization property is illustrated by
x^2 + 2x + 1 -> (x+1)(x+2)(x+1) = x^3 + 4x^2 + 5x + 2.

Mathematica

s =.; s[1] = {1};
Prepend[Table[s[z] = Table[SymmetricPolynomial
 [k, s[z - 1]], {k, 0, z - 1}], {z, 2, 7}], s[1]]
% // TableForm (* A203300 triangle *)
%% // Flatten  (* A203300 sequence *)
(* Peter J. C. Moses, Dec 30 2011 *)

Crossrefs

Cf. A203301.

Sequence in context: A118686 A355540 A102610 * A134172 A208061 A078047

Adjacent sequences: A203297 A203298 A203299 * A203301 A203302 A203303

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 31 2011

Status

approved