cp's OEIS Frontend

A203006 (n-1)-st elementary symmetric function of the first n Fibonacci numbers.

%I A203006 #17 Aug 18 2024 20:11:27
%S A203006 1,2,5,17,91,758,10094,215094,7378716,408057060,36439600740,
%T A203006 5258207000160,1226732478115680,462844011818878560,
%U A203006 282472779283129656000,278884771717353348456000,445462025196173918554440000,1151206495594319717393795136000
%N A203006 (n-1)-st elementary symmetric function of the first n Fibonacci numbers.
%C A203006 From _R. J. Mathar_, Oct 01 2016: (Start)
%C A203006 The k-th elementary symmetric functions of the integers F(j), j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
%C A203006 1;
%C A203006 1, 1;
%C A203006 1, 2, 1;
%C A203006 1, 4, 5, 2;
%C A203006 1, 7, 17, 17, 6;
%C A203006 which is the unsigned version of A158472. This here is the first subdiagonal. The diagonal seems to be A003266. The 2nd column is A000071, the 3rd A190173, the 4th A213787. (End)
%H A203006 Robert Israel, <a href="/A203006/b203006.txt">Table of n, a(n) for n = 1..98</a>
%e A203006 0th elementary symmetric function: 1
%e A203006 1st e.s.f. of {1,1}: 1+1=2
%e A203006 2nd e.s.f. of {1,1,2}: 1*1+1*2+2*2=5
%p A203006 f:= proc(n) local x,P,i;
%p A203006 P:= mul(x+combinat:-fibonacci(i),i=1..n);
%p A203006 coeff(P,x,1)
%p A203006 end proc:
%p A203006 map(f, [$1..20]); # _Robert Israel_, Aug 18 2024
%t A203006 f[k_] := Fibonacci[k]; t[n_] := Table[f[k], {k, 1, n}]
%t A203006 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203006 Table[a[n], {n, 1, 18}]  (* A203006 *)
%Y A203006 Cf. A000045.
%K A203006 nonn
%O A203006 1,2
%A A203006 _Clark Kimberling_, Dec 29 2011