A380296 Number of transfer systems for the Dihedral group of order 2p^n, with p an odd prime.

2, 9, 56, 416, 3457, 31063, 295834, 2948082, 30471080, 324580196, 3546142551, 39588702271, 450277384320, 5205233568669, 61037153047708, 724817556942798, 8704492269996637, 105591602247646356, 1292561576650363350, 15952703801125660022, 198359915092340815084

Offset

0,1

Comments

Balchin-MacBrough-Ormsby give a complicated recurrence relation for this sequence, but do not produce a closed form.

Links

Ben Spitz, Table of n, a(n) for n = 0..50

Scott Balchin, Code to generate the sequence, denoted |L(n)|.

S. Balchin, E. MacBrough, and K. Ormsby, The combinatorics of N_oo operads for C_{qp^n} and D_{p^n}, arXiv:2209.06992 [math.AT], 2022-2024.

S. Balchin, E. MacBrough, and K. Ormsby, The combinatorics of N_oo operads for C_{qp^n} and D_{p^n}, Glasgow Mathematical Journal, First View, pp. 1-17.

Formula

a(n) = sum of L(n,k,l,a,b) ranging over 1 <= k,l <= n+1, 0 <= a <= 1, 0 <= b <= n, where

L(n,k,l,0,0) = sum of L(n-1,k',l-1,a,b) ranging over k-1 <= k' <= n, 0 <= a <= 1, 0 <= b <= n-1

L(n,k,l,1,n) = sum of L(n-1,k-1,l',a,b) ranging over l-1 <= l' <= n, 0 <= a <= 1, 0 <= b <= n-1

For b>0, L(n,k,l,0,b) = sum of T(b-1,i)*L(n-b,k-i,l,0,0) ranging over 0 <= i <= k

For b<n, L(n,k,l,1,b) = 0

Otherwise, L(n,k,l,a,b) = 0,

where in the above T(x,y) = (2(2y + 1)!(4x - 2y + 3)!)/((y - 1)!(y + 1)!(x - y + 1)!(3x - y + 4)!).

Crossrefs

Sequence in context: A081004 A198953 A212392 * A186262 A138740 A276370

Adjacent sequences: A380293 A380294 A380295 * A380297 A380298 A380299

Keyword

nonn

Author

Ben Spitz, Jan 22 2025

Status

approved