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Arises in combinatorial approach to the power of 2 in the number of involutions.
1

%I #17 May 07 2023 10:07:58

%S 1,1,1,2,2,6,8,26,41,145,253,978,1858,7726,15796,69878,152219,711243,

%T 1638323,8039510,19467494,99862594,252998224,1351486758,3568259503,

%U 19786100599,54263159347,311482467134,884834059454,5245588599330,15397757661092

%N Arises in combinatorial approach to the power of 2 in the number of involutions.

%C Table 1 of Kim and Kim.

%H Andrew Howroyd, <a href="/A157253/b157253.txt">Table of n, a(n) for n = 0..200</a>

%H Dongsu Kim and Jang Soo Kim, <a href="http://arxiv.org/abs/0902.4311">A combinatorial approach to the power of 2 in the number of involutions</a>, arXiv:0902.4311 [math.CO], 2009-2010.

%F a(2*n) = a(2*n-2) + (n-1)*a(2*n-3) + 2*binomial(n-1, 2)*a(2*n-4) + 3*binomial(n-1, 3)*a(2*n-8); a(2*n+1) = a(2*n) + n*a(2*n-1). See eqn. 5 and 6 for g_n(1,1) in Kim and Kim reference. - _Andrew Howroyd_, May 07 2023

%o (PARI)

%o seq(n)={my(a=vector(n+1)); a[1]=a[2]=1; for(n=2, n, a[n+1] = if(n%2==0, a[n-1] + if(n>=3, (n/2-1)*a[n-2]) + if(n>=4, 2*binomial(n/2-1,2)*a[n-3]) + if(n>=8, 3*binomial(n/2-1,3)*a[n-7]), a[n] + (n-1)*a[n-1]/2)); a} \\ _Andrew Howroyd_, May 06 2023

%K nonn

%O 0,4

%A _Jonathan Vos Post_, Feb 25 2009

%E a(7)-a(21) from Kim paper by _Michel Marcus_, Sep 04 2014

%E Missing a(19) inserted and more terms from _Andrew Howroyd_, May 06 2023