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A203156 (n-1)-st elementary symmetric function of {4,9,16,25,..., (n+1)^2}.

%I A203156 #38 Dec 16 2024 16:50:29
%S A203156 1,13,244,6676,254736,13000464,857431296,71077637376,7239445632000,
%T A203156 889141110912000,129629670893568000,22136856913815552000,
%U A203156 4377599743151480832000,992559996665635184640000,255805371399126806691840000
%N A203156 (n-1)-st elementary symmetric function of {4,9,16,25,..., (n+1)^2}.
%H A203156 Robert Israel, <a href="/A203156/b203156.txt">Table of n, a(n) for n = 1..251</a>
%F A203156 a(n) = gamma(2 + n)^2*(Pi^2/6 - 1 - digamma^(1)(2 + n)), where gamma(x) is the gamma function and digamma^(n)(x) is the n-th derivative of the digamma function. - _Markus Bindhammer_, Nov 26 2017
%e A203156 Let esf abbreviate "elementary symmetric function". Then
%e A203156 0th esf of {4}:  1;
%e A203156 1st esf of {4,9}:  4 + 9 = 13;
%e A203156 2nd esf of {4,9,16}:  4*9 + 4*16 + 9*16 = 244.
%p A203156 f:= proc(n) local i,x; coeff(mul(x+i^2,i=2..n+1),x,1) end proc:
%p A203156 map(f, [$1..26]); # _Robert Israel_, Dec 16 2024
%t A203156 f[k_] := (k + 1)^2; t[n_] := Table[f[k], {k, 1, n}]
%t A203156 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203156 Table[a[n], {n, 1, 22}] (* A203156 *)
%Y A203156 Cf. A066989.
%K A203156 nonn
%O A203156 1,2
%A A203156 _Clark Kimberling_, Dec 29 2011