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A024449 4th elementary symmetric function of the first n+3 primes.

%I A024449 #13 Mar 19 2020 16:00:55
%S A024449 210,2927,20581,107315,414849,1376640,4224150,11063618,27395788,
%T A024449 62364155,129081579,252768753,480307611,885449578,1541654028,
%U A024449 2623783892,4318819858,6832984023,10644660237,16195499543,24304992465,36231495836,52916319106,75433702422
%N A024449 4th elementary symmetric function of the first n+3 primes.
%H A024449 Alois P. Heinz, <a href="/A024449/b024449.txt">Table of n, a(n) for n = 1..10000</a>
%p A024449 SymmPolyn := proc(L::list,n::integer)
%p A024449     local c,a,sel;
%p A024449     a :=0 ;
%p A024449     sel := combinat[choose](nops(L),n) ;
%p A024449     for c in sel do
%p A024449         a := a+mul(L[e],e=c) ;
%p A024449     end do:
%p A024449     a;
%p A024449 end proc:
%p A024449 A024449 := proc(n)
%p A024449     [seq(ithprime(k),k=1..n+3)] ;
%p A024449     SymmPolyn(%,4) ;
%p A024449 end proc: # _R. J. Mathar_, Sep 23 2016
%p A024449 # second Maple program:
%p A024449 b:= proc(n) option remember; convert(series(`if`(n=0, 1,
%p A024449       b(n-1)*(ithprime(n)*x+1)), x, 5), polynom)
%p A024449     end:
%p A024449 a:= n-> coeff(b(n+3), x, 4):
%p A024449 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 06 2019
%t A024449 b[n_] := b[n] = Series[If[n == 0, 1, b[n - 1] (Prime[n] x + 1)], {x, 0, 5}] // Normal;
%t A024449 a[n_] := Coefficient[b[n + 3], x, 4];
%t A024449 a /@ Range[24] (* _Jean-François Alcover_, Mar 19 2020, after _Alois P. Heinz_ *)
%o A024449 (PARI) e4(v)=sum(i=1,#v-3,v[i]*sum(j=i+1,#v-2,v[j]*sum(k=j+1,#v-1,v[k]*vecsum(v[k+1..#v]))))
%o A024449 a(n)=e4(primes(n)) \\ _Charles R Greathouse IV_, Jun 15 2015
%K A024449 nonn
%O A024449 1,1
%A A024449 _Clark Kimberling_