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a(n) = (k-1)st elementary symmetric function of C(n,0), C(n,1), ..., C(n,k), where k = floor( n/2 ).
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%I #12 Sep 08 2019 12:48:03

%S 1,1,11,16,551,1190,178024,564678,410606100,1876011225,6915255136416,

%T 44675417804160,847468391006481244,7637169791538787500,

%U 749927054569389785088000,9345619999880270191554560,4766524174302701575265292220416,81712716729371439637617531305856

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

%H Andrew Howroyd, <a href="/A025141/b025141.txt">Table of n, a(n) for n = 2..50</a>

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

%p seq(a(n), n=2..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]]];

%t a /@ Range[2, 19] (* _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)={if(n>=2, ESym(binomial(n)[1..1+n\2])[n\2])} \\ _Andrew Howroyd_, Dec 19 2018

%K nonn

%O 2,3

%A _Clark Kimberling_

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