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(n-1)-st elementary symmetric function of {1,1,2,2,3,3,4,4,5,5,...,Floor[(n+1)/2]}.
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%I #16 Nov 28 2017 11:35:59

%S 1,2,5,12,40,132,564,2400,12576,65760,408960,2540160,18299520,

%T 131725440,1079205120,8836853760,81157386240,745047797760,

%U 7582159872000,77138417664000,861690783744000,9623448705024000,117074735382528000

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

%C Column 3 of A246117. - _Peter Bala_, Aug 15 2014

%C From _R. J. Mathar_, Oct 01 2016 (Start):

%C 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 1

%C 1 1

%C 1 2 1

%C 1 4 5 2

%C 1 6 13 12 4

%C 1 9 31 51 40 12

%C 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 Let esf abbreviate "elementary symmetric function". Then

%e 0th esf of {2}: 1;

%e 1st esf of {1,1}: 1+1=2;

%e 2nd esf of {1,1,2} is 1*1+1*2+1*2=5.

%t f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[n - 1, t[n]]

%t Table[a[n], {n, 1, 22}] (* A203151 *)

%Y Cf. A203152, A246117.

%K nonn

%O 1,2

%A _Clark Kimberling_, Dec 29 2011