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A203154 (n-1)-st elementary symmetric function of {2, 3, 3, 4, 4, 5, 5,...,Floor[(n+4)/2]}.

%I A203154 #18 Nov 27 2017 16:50:04
%S A203154 1,5,21,102,480,2688,14880,96480,622080,4613760,34110720,285586560,
%T A203154 2386298880,22289541120,207921530880,2145056256000,22108972032000,
%U A203154 249782787072000,2820035699712000,34637103857664000,425205351825408000
%N A203154 (n-1)-st elementary symmetric function of {2, 3, 3, 4, 4, 5, 5,...,Floor[(n+4)/2]}.
%H A203154 Robert Israel, <a href="/A203154/b203154.txt">Table of n, a(n) for n = 1..502</a>
%e A203154 Let esf abbreviate "elementary symmetric function". Then
%e A203154 0th esf of {2}:  1
%e A203154 1st esf of {2,3}:  2+3=5;
%e A203154 2nd esf of {2,3,3} is 2*3+2*3+3*3=21.
%p A203154 SymmPolyn := proc(L::list,n::integer)
%p A203154     local c,a,sel;
%p A203154     a :=0 ;
%p A203154     sel := combinat[choose](nops(L),n) ;
%p A203154     for c in sel do
%p A203154         a := a+mul(L[e],e=c) ;
%p A203154     end do:
%p A203154     a;
%p A203154 end proc:
%p A203154 A203154 := proc(n)
%p A203154     [seq(floor((k+4)/2),k=1..n)] ;
%p A203154     SymmPolyn(%,n-1);
%p A203154 end proc:  # _R. J. Mathar_, Sep 23 2016
%p A203154 # second Maple program:
%p A203154 f:=  proc(n) local L,x;
%p A203154   if n::odd then L:= `*`(x+2,seq((x+i)$2, i=3..2+n/2))
%p A203154   else L:= `*`(seq((x+i)*(x+i+1),i=2..1+n/2))
%p A203154   fi;
%p A203154   coeff(L,x,1);
%p A203154 end proc:
%p A203154 map(f, [$1..50]); # _Robert Israel_, Nov 27 2017
%t A203154 f[k_] := Floor[(k + 4)/2]; t[n_] := Table[f[k], {k, 1, n}]
%t A203154 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203154 Table[a[n], {n, 1, 22}] (* A203154 *)
%Y A203154 Cf. A203152, A203153, A203155.
%K A203154 nonn
%O A203154 1,2
%A A203154 _Clark Kimberling_, Dec 29 2011