%I #31 Oct 01 2022 00:39:49

%S 1,1,2,4,13,41,176,722,3774,18958,116302,687182,4812226,32541874,

%T 255274826,1938568634,16798483589,141220228149,1337121257864,

%U 12305678519102,126208299343263,1260257489267963,13901541357573146,149520289244078172,1763398965493327476

%N The number of evenly tagged partitions: partitions of n elements together with an involution defined on the set of classes which has at most one fixed point, such that a class and its image have the same number of elements.

%C a(n) is also the number of subspaces of R^n given by coordinate equalities of the form x_i = x_j, x_i = -x_j and x_i = 0, that are orthogonal to the vector of all 1's.

%C These subspaces play an important role in the field of network dynamical systems, where they correspond to anti-synchronization. That is, they capture the phenomenon where different cells in a network show the same or opposite behavior.

%H James W. Swift, <a href="/A355194/b355194.txt">Table of n, a(n) for n = 0..500</a>

%H Eddie Nijholt, Nándor Sieben, and James W. Swift, <a href="https://arxiv.org/abs/2206.00094">Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks</a>, arXiv:2206.00094 [math.DS], 2022.

%F E.g.f.: exp(I_0(2z)/2 - 1/2 + z), where I_0 is the modified Bessel function of the first kind.

%F a(0) = 1, a(n+1) = a(n) + Sum_{k>=0, m>=0, k + 2m + 1 = n} (n!/(k!*m!*(n-k-m)!)*a(k).

%e The set {1,2} has the trivial partition consisting only of the class {1,2}, together with the involution that sends {1,2} to itself. There is also the singleton partition with classes {1} and {2}, together with the involution that maps {1} to {2} and vice versa. The other involution on the two classes {1} and {2} has two fixed points and is therefore not counted, Hence, we find a(2) = 2.

%e Alternatively, in R^2 we have two subspaces given by coordinate equalities of the form x_i = x_j, x_i = -x_j and x_i = 0, that are orthogonal to the vector (1,1). These are the zero-space and the subspace given by x_1 = -x_2. In R^3 we find the zero-space and the three subspaces given by x_i = -x_j, x_k = 0 for {i,j,k} = {1,2,3}. This shows that a(3) = 4.

%t nMax = 20; CoefficientList[ Series[Exp[BesselI[0, 2 x]/2 - 1/2 + x], {x, 0, nMax}], x] * Range[0, nMax]!

%Y Cf. A350291.

%K nonn

%O 0,3

%A _Eddie Nijholt_, Jun 23 2022