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

%I A380296 #28 Jan 27 2025 15:39:43
%S A380296 2,9,56,416,3457,31063,295834,2948082,30471080,324580196,3546142551,
%T A380296 39588702271,450277384320,5205233568669,61037153047708,
%U A380296 724817556942798,8704492269996637,105591602247646356,1292561576650363350,15952703801125660022,198359915092340815084
%N A380296 Number of transfer systems for the Dihedral group of order 2p^n, with p an odd prime.
%C A380296 Balchin-MacBrough-Ormsby give a complicated recurrence relation for this sequence, but do not produce a closed form.
%H A380296 Ben Spitz, <a href="/A380296/b380296.txt">Table of n, a(n) for n = 0..50</a>
%H A380296 Scott Balchin, <a href="https://github.com/bifibrant/recursion/">Code to generate the sequence, denoted |L(n)|</a>.
%H A380296 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 A380296 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 A380296 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 A380296 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 A380296 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 A380296 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 A380296 For b<n, L(n,k,l,1,b) = 0
%F A380296 Otherwise, L(n,k,l,a,b) = 0,
%F A380296 where in the above T(x,y) = (2(2y + 1)!(4x - 2y + 3)!)/((y - 1)!(y + 1)!(x - y + 1)!(3x - y + 4)!).
%K A380296 nonn
%O A380296 0,1
%A A380296 _Ben Spitz_, Jan 22 2025