A208655 Number of palindromic paths starting with 1 through the subset array of {1,2,...,n}; see Comments.

1, 1, 6, 36, 2400, 90000, 55566000, 13553164800, 72267023646720, 117595223746560000, 5219033393851200000000, 57636380373471744768000000, 20810790197418148654769602560000, 1578992018570629416640340512656998400

Offset

1,3

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.

Links

Table of n, a(n) for n=1..14.

Example

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 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.

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 *)

Crossrefs

Cf. A208650, A208654.

Sequence in context: A061234 A061584 A197191 * A202781 A067213 A173604

Adjacent sequences: A208652 A208653 A208654 * A208656 A208657 A208658

Keyword

nonn

Author

Clark Kimberling, Mar 02 2012

Status

approved