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Number of noncrossing anti-commutator friendly partitions on {1,2,...,2n}.
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%I #40 Mar 06 2022 09:39:02

%S 1,5,22,117,678,4162,26588,174925,1177158,8064854,56062804,394443458,

%T 2803490524,20098913252,145175116408,1055463627197,7717664983366,

%U 56720231324046,418757618733092,3104269959560566

%N Number of noncrossing anti-commutator friendly partitions on {1,2,...,2n}.

%C Let n be a positive integer, let sigma be a noncrossing partition on {1,2,...,2n}. Consider the set OuterMax(sigma) := {max(W); W is an outer block of sigma}.

%C We say that sigma is anti-commutator friendly when it satisfies:

%C 1) OuterMax(sigma) is a subset of the union of {1,3,...,2n-1} and {2n}.

%C 2) For every j in {1, 3,...,2n-1} \ OuterMax(sigma), one has depth(j) is not equal to depth(j + 1), where depth(j) stands for the depth of the block of sigma which contains the number j.

%C For example for n=1 only partition {{1},{2}} is anti-commutator friendly.

%C Multiplied by two, gives sequence of Boolean cumulants of ab+ba, for a,b freely independent both distributed (delta_0+delta_2)/2. See M. Fevrier et al. Proposition 6.11.

%H M. Fevrier, M. Mastnak, A. Nica and K. Szpojankowski, <a href="https://arxiv.org/abs/1907.10842">Using Boolean cumulants to study multiplication and anti-commutators of free random variables</a>, arXiv:1907.10842 [math.OA], 2019.

%F G.f.: 1/2 - sqrt( (1 - 8*z)*(1 - 2*z-sqrt(1-8*z))/(8*z) ).

%F G.f.: 1/2 - sqrt( 1/4 + 3*z - 4*z*g(z) ), where g(z) is the g.f. of A000257.

%F a(n) ~ sqrt(3) * 2^(3*n-2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 29 2020

%F D-finite with recurrence n*(2*n+1)*a(n) +2*(-11*n^2+17*n-3)*a(n-1) +24*(-n^2+7*n-8)*a(n-2) +32*(16*n^2-109*n+186)*a(n-3) +256*(n-4)*(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Mar 06 2022

%t Rest[CoefficientList[Series[1/2 - Sqrt[(1 - 8*x)*(1 - Sqrt[1 - 8*x] - 2*x)/(8*x)], {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Jan 29 2020 *)

%o (PARI) seq(n)={Vec(1/2 - sqrt((1 - 8*x)*(1-2*x-sqrt(1 - 8*x + O(x^2*x^n)))/(8*x)))} \\ _Andrew Howroyd_, Jan 28 2020

%Y Cf. A000257.

%K nonn,easy

%O 1,2

%A _Kamil Szpojankowski_, Jan 28 2020