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

1, 2, 18, 144, 12000, 540000, 388962000, 108425318400, 650403212820480, 1175952237465600000, 57409367332363200000000, 691636564481660937216000000, 270540272566435932512004833280000

Offset: 1

Author

Clark Kimberling, Mar 02 2012

Keywords

• nonn

Comments

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.

Examples

			For n=4, write
row 1:  1,2,3,4
row 2:  1,2; 1,3; 1,4; 2,3; 2,4; 3;4
row 3:  1,2,3; 1,2,4; 1,3,4; 2,3,4
row 4:  1,2,3,4
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.
		

Crossrefs

Cf. A208650, A208655.

Programs

Mathematica

m[n_] := Floor[(n + 1)/2]; z = 21;
g[n_] := Product[i*Binomial[n, i], {i, 1, m[n]}]
h[n_] := Product[Binomial[n - 1, i], {i, m[n], n - 1}]
Table[g[n], {n, 1, z}]   (* A208652 *)
Table[h[n], {n, 1, z}]   (* A208653 *)
Table[g[n] h[n], {n, 1, 2 z/3}]   (* A208654 *)
Table[g[n] h[n]/n, {n, 1, 2 z/3}] (* A208655 *)