A025140 - OEIS

%I #10 Sep 08 2019 13:04:19

%S 1,1,3,4,34,65,2310,6272,1047424,3973536,3255714000,17050908600,

%T 69896096519040,501898385570868,10413963558878928048,

%U 102159685299672000000,10820560943026950635520000,144743469304894583259136000,78786195510356832343493745377280

%N a(n) = floor(n/2)-th elementary symmetric function of C(n,0), C(n,1), ..., C(n, floor(n/2)).

%H Andrew Howroyd, <a href="/A025140/b025140.txt">Table of n, a(n) for n = 0..50</a>

%p a:= n-> (k-> coeff(mul(binomial(n, i)*x+1, i=0..k), x, k))(iquo(n, 2)):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 08 2019

%t ESym[u_] := Module[{v, t}, v = Table[0, {Length[u] + 1}]; v[[1]] = 1; For[i = 1, i <= Length[u], i++, t = u[[i]]; For[j = i, j >= 1, j--, v[[j + 1]] += v[[j]]*t]]; v];

%t a[n_] := ESym[Table[Binomial[n, k], {k, 0, Floor[n/2]}]][[Floor[n/2] + 1]];

%t a /@ Range[0, 18] (* _Jean-François Alcover_, Sep 08 2019, from PARI *)

%o (PARI)

%o ESym(u)={my(v=vector(#u+1)); v[1]=1; for(i=1, #u, my(t=u[i]); forstep(j=i, 1,-1, v[j+1]+=v[j]*t)); v}

%o a(n)={ESym(binomial(n)[1..1+n\2])[n\2+1]} \\ _Andrew Howroyd_, Dec 19 2018

%K nonn

%O 0,3

%A _Clark Kimberling_

%E Terms a(14) and beyond from _Andrew Howroyd_, Dec 19 2018