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A203151 (n-1)-st elementary symmetric function of {1,1,2,2,3,3,4,4,5,5,...,Floor[(n+1)/2]}.

%I A203151 #16 Nov 28 2017 11:35:59
%S A203151 1,2,5,12,40,132,564,2400,12576,65760,408960,2540160,18299520,
%T A203151 131725440,1079205120,8836853760,81157386240,745047797760,
%U A203151 7582159872000,77138417664000,861690783744000,9623448705024000,117074735382528000
%N A203151 (n-1)-st elementary symmetric function of {1,1,2,2,3,3,4,4,5,5,...,Floor[(n+1)/2]}.
%C A203151 Column 3 of A246117. - _Peter Bala_, Aug 15 2014
%C A203151 From _R. J. Mathar_, Oct 01 2016 (Start):
%C A203151 The k-th elementary symmetric functions of the repeated integers 1,1,2,2,..[(n+1)/2], form a triangle T(n,k), 0<=k<=n, n>=0:
%C A203151 1
%C A203151 1 1
%C A203151 1 2 1
%C A203151 1 4 5 2
%C A203151 1 6 13 12 4
%C A203151 1 9 31 51 40 12
%C A203151 which is a row-reversed version of A246117.  This here is the first subdiagonal. The diagonal is A010551. The 2nd column is A002620, the 3rd A203246. (End)
%e A203151 Let esf abbreviate "elementary symmetric function".  Then
%e A203151 0th esf of {2}:  1;
%e A203151 1st esf of {1,1}:  1+1=2;
%e A203151 2nd esf of {1,1,2} is 1*1+1*2+1*2=5.
%t A203151 f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]
%t A203151 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203151 Table[a[n], {n, 1, 22}] (* A203151 *)
%Y A203151 Cf. A203152, A246117.
%K A203151 nonn
%O A203151 1,2
%A A203151 _Clark Kimberling_, Dec 29 2011