A195186 Number of palindromic double occurrence words of length 2n.

1, 2, 6, 20, 72, 290, 1198, 5452, 25176, 125874, 637926, 3448708, 18919048, 109412210, 642798510, 3945170012, 24614491704, 159328958690, 1048645656646, 7122719571700, 49185991168968, 349097516604738, 2518145666958126, 18609525157571692, 139704193446510616

Offset: 1

Author

N. J. A. Sloane, Sep 10 2011

Keywords

• nonn

Links

• Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.

Programs

Maple

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:
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:
seq(A195186(n),n=1..20) ; # R. J. Mathar, Sep 12 2011

Mathematica

b[n_] := Sum[Binomial[k, n - k]*(n!/k!), {k, 0, n}];
a[1] = 1; a[n_] := b[n] - Sum[a[n - 2*k]*(2*k - 1)!!, {k, 1, n/2}];
Array[a, 20] (* Jean-François Alcover, Nov 29 2017, after R. J. Mathar *)

Formula

Theorem 3.3 of Burns-Muche gives a recurrence.