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A203300 Self-generating triangle based on symmetric functions. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A198895 A118686 A102610 * A134172 A208061 A078047

Adjacent sequences:  A203297 A203298 A203299 * A203301 A203302 A203303

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Dec 31 2011

STATUS

approved

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Last modified August 21 19:52 EDT 2017. Contains 290906 sequences.