A380296 - OEIS

%I #28 Jan 27 2025 15:39:43

%S 2,9,56,416,3457,31063,295834,2948082,30471080,324580196,3546142551,

%T 39588702271,450277384320,5205233568669,61037153047708,

%U 724817556942798,8704492269996637,105591602247646356,1292561576650363350,15952703801125660022,198359915092340815084

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

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

%H Ben Spitz, <a href="/A380296/b380296.txt">Table of n, a(n) for n = 0..50</a>

%H Scott Balchin, <a href="https://github.com/bifibrant/recursion/">Code to generate the sequence, denoted |L(n)|</a>.

%H S. Balchin, E. MacBrough, and K. Ormsby, <a href="https://arxiv.org/abs/2209.06992">The combinatorics of N_oo operads for C_{qp^n} and D_{p^n}</a>, arXiv:2209.06992 [math.AT], 2022-2024.

%H S. Balchin, E. MacBrough, and K. Ormsby, <a href="https://doi.org/10.1017/S0017089524000211">The combinatorics of N_oo operads for C_{qp^n} and D_{p^n}</a>, Glasgow Mathematical Journal, First View, pp. 1-17.

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

%F 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

%F 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

%F 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

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

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

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

%K nonn

%O 0,1

%A _Ben Spitz_, Jan 22 2025