A203154 - OEIS

%I #18 Nov 27 2017 16:50:04

%S 1,5,21,102,480,2688,14880,96480,622080,4613760,34110720,285586560,

%T 2386298880,22289541120,207921530880,2145056256000,22108972032000,

%U 249782787072000,2820035699712000,34637103857664000,425205351825408000

%N (n-1)-st elementary symmetric function of {2, 3, 3, 4, 4, 5, 5,...,Floor[(n+4)/2]}.

%H Robert Israel, <a href="/A203154/b203154.txt">Table of n, a(n) for n = 1..502</a>

%e Let esf abbreviate "elementary symmetric function". Then

%e 0th esf of {2}:  1

%e 1st esf of {2,3}:  2+3=5;

%e 2nd esf of {2,3,3} is 2*3+2*3+3*3=21.

%p SymmPolyn := proc(L::list,n::integer)

%p     local c,a,sel;

%p     a :=0 ;

%p     sel := combinat[choose](nops(L),n) ;

%p     for c in sel do

%p         a := a+mul(L[e],e=c) ;

%p     end do:

%p     a;

%p end proc:

%p A203154 := proc(n)

%p     [seq(floor((k+4)/2),k=1..n)] ;

%p     SymmPolyn(%,n-1);

%p end proc:  # _R. J. Mathar_, Sep 23 2016

%p # second Maple program:

%p f:=  proc(n) local L,x;

%p   if n::odd then L:= `*`(x+2,seq((x+i)$2, i=3..2+n/2))

%p   else L:= `*`(seq((x+i)*(x+i+1),i=2..1+n/2))

%p   fi;

%p   coeff(L,x,1);

%p end proc:

%p map(f, [$1..50]); # _Robert Israel_, Nov 27 2017

%t f[k_] := Floor[(k + 4)/2]; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[n - 1, t[n]]

%t Table[a[n], {n, 1, 22}] (* A203154 *)

%Y Cf. A203152, A203153, A203155.

%K nonn

%O 1,2

%A _Clark Kimberling_, Dec 29 2011