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Number of palindromic double occurrence words of length 2n.
1

%I #18 Nov 29 2017 16:57:06

%S 1,2,6,20,72,290,1198,5452,25176,125874,637926,3448708,18919048,

%T 109412210,642798510,3945170012,24614491704,159328958690,

%U 1048645656646,7122719571700,49185991168968,349097516604738,2518145666958126,18609525157571692,139704193446510616

%N Number of palindromic double occurrence words of length 2n.

%H Jonathan Burns and Tilahun Muche, <a href="http://arxiv.org/abs/1105.2926">Counting Irreducible Double Occurrence Words</a>, arXiv preprint arXiv:1105.2926 [math.CO], 2011.

%F Theorem 3.3 of Burns-Muche gives a recurrence.

%p A047974 := proc(n) option remember; if n= 1 then 1; elif n=2 then 3; else procname(n-1)+2*(n-1)*procname(n-2) ; end if; end proc:

%p A195186 := proc(n) if n <= 1 then 1; else A047974(n)-add(procname(n-2*k)*doublefactorial(2*k-1),k=1..floor(n/2)) ; end if; end proc:

%p seq(A195186(n),n=1..20) ; # _R. J. Mathar_, Sep 12 2011

%t b[n_] := Sum[Binomial[k, n - k]*(n!/k!), {k, 0, n}];

%t a[1] = 1; a[n_] := b[n] - Sum[a[n - 2*k]*(2*k - 1)!!, {k, 1, n/2}];

%t Array[a, 20] (* _Jean-François Alcover_, Nov 29 2017, after _R. J. Mathar_ *)

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Sep 10 2011