%I #5 Mar 30 2012 18:58:13
%S 1,1,6,36,2400,90000,55566000,13553164800,72267023646720,
%T 117595223746560000,5219033393851200000000,57636380373471744768000000,
%U 20810790197418148654769602560000,1578992018570629416640340512656998400
%N Number of palindromic paths starting with 1 through the subset array of {1,2,...,n}; see Comments.
%C 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 kelement subsets of {1,2,...,n}. See A208650 for a discussion and guide to related sequences.
%e For n=4, write
%e row 1: 1,2,3,4
%e row 2: 1,2; 1,3; 1,4; 2,3; 2,4; 3;4
%e row 3: 1,2,3; 1,2,4; 1,3,4; 2,3,4
%e row 4: 1,2,3,4
%e To form a palindromic path of length 4 and starting with 1, there is 1 way 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. Thus, a(4)=1*12*3*1=36.
%t m[n_] := Floor[(n + 1)/2]; z = 21;
%t g[n_] := Product[i*Binomial[n, i], {i, 1, m[n]}]
%t h[n_] := Product[Binomial[n  1, i], {i, m[n], n  1}]
%t Table[g[n], {n, 1, z}] (* A208652 *)
%t Table[h[n], {n, 1, z}] (* A208653 *)
%t Table[g[n] h[n], {n, 1, 2 z/3}] (* A208654 *)
%t Table[g[n] h[n]/n, {n, 1, 2 z/3}] (* A208655 *)
%Y Cf. A208650, A208654.
%K nonn
%O 1,3
%A _Clark Kimberling_, Mar 02 2012
