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A208654 Number of palindromic paths through the subset array of {1,2,...,n}; see Comments.

%I A208654 #5 Mar 30 2012 18:58:13
%S A208654 1,2,18,144,12000,540000,388962000,108425318400,650403212820480,
%T A208654 1175952237465600000,57409367332363200000000,
%U A208654 691636564481660937216000000,270540272566435932512004833280000
%N A208654 Number of palindromic paths through the subset array of {1,2,...,n}; see Comments.
%C A208654 A palindromic path through the subset array of {1,2,...,n} is essentially a palindrome using numbers i from {1,2,...n}, where the number of times i can be used in position k equals the multiplicity of i in the multiset of numbers in the k-element subsets of {1,2,...,n}.  See A208650 for a discussion and guide to related sequences.
%e A208654 For n=4, write
%e A208654 row 1:  1,2,3,4
%e A208654 row 2:  1,2; 1,3; 1,4; 2,3; 2,4; 3;4
%e A208654 row 3:  1,2,3; 1,2,4; 1,3,4; 2,3,4
%e A208654 row 4:  1,2,3,4
%e A208654 To form a palindromic path of length 4, there are 4 ways to choose 1st term from row 1, then 12 ways to choose 2nd term from row 2, then 3 ways to choose 3rd term, then 1 way to finish, so that a(4)=4*12*3*1=144.
%t A208654 m[n_] := Floor[(n + 1)/2]; z = 21;
%t A208654 g[n_] := Product[i*Binomial[n, i], {i, 1, m[n]}]
%t A208654 h[n_] := Product[Binomial[n - 1, i], {i, m[n], n - 1}]
%t A208654 Table[g[n], {n, 1, z}]   (* A208652 *)
%t A208654 Table[h[n], {n, 1, z}]   (* A208653 *)
%t A208654 Table[g[n] h[n], {n, 1, 2 z/3}]   (* A208654 *)
%t A208654 Table[g[n] h[n]/n, {n, 1, 2 z/3}] (* A208655 *)
%Y A208654 Cf. A208650, A208655.
%K A208654 nonn
%O A208654 1,2
%A A208654 _Clark Kimberling_, Mar 02 2012