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%I #34 May 18 2019 02:05:02
%S 1,1,3,8,27,104,436,1930,8871,41916,202300,992942,4940912,24867870,
%T 126371426,647494746,3341341155,17350565376,90593056624,475333630402,
%U 2504959102224,13252904123786,70366654738470,374824160997086
%N Number of signed weighted Euler trees with total weight n (associated to even switching classes of matrices of order 2n).
%H Vincenzo Librandi, <a href="/A110886/b110886.txt">Table of n, a(n) for n = 0..300</a>
%H R. Bacher and D. Garber, <a href="http://www.ams.org/mathscinet-getitem?mr=2326942">Spindle-configurations of skew lines</a>, Geom. Topol 11 (2007) 1049.
%F G.f.: ( 3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)) ) / (2*(1-z)).
%F a(n) = 2 + Sum_{k=1..n-1} a(n-k)*a(k). - _Benoit Cloitre_, Jul 27 2008
%F Recurrence: n*a(n) = 2*(3*n-4)*a(n-1) - (n+2)*a(n-2) - 2*(2*n-7)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ sqrt(41-3*sqrt(41))*((5+sqrt(41))/2)^n/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) = Sum_{k=1..n} (binomial(2*k-2, k-1)*Sum_{i=0..n-k} binomial(k, n-k-i)*binomial(k+i-1, k-1)/k), n > 0, a(0)=1. - _Vladimir Kruchinin_, Jan 24 2013
%F a(n+1) starting (1, 3, ...) = (first n terms) dot product (first n difference terms), added to a(n). - _Gary W. Adamson_, May 20 2013
%e a(5) = 104. (1, 3, 8, 27) dot (1, 2, 5, 19) = 77; then 104 = a(4) + 77 = 27 + 77.
%p G:=(3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)))/2/(1-z): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27); # _Emeric Deutsch_, Dec 31 2006
%t CoefficientList[Series[(3*(1-x)-Sqrt[(1-x)*(1-5*x-4*x^2)])/2/(1-x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%t a[n_] := Sum[(Binomial[2*k-2, k-1]*Sum[Binomial[k, n-k-i]*Binomial[k+i-1, k-1], {i, 0, n-k}])/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jan 24 2013, after _Vladimir Kruchinin_ *)
%o (Maxima) a(n):=sum((binomial(2*k-2,k-1)*sum(binomial(k,n-k-i)*binomial(k+i-1,k-1),i,0,n-k))/k,k,1,n); /* _Vladimir Kruchinin_, Jan 24 2013 */
%o (PARI)
%o N = 66; x = 'x + O('x^N);
%o gf = ( 3*(1-x)-sqrt((1-x)*(1-5*x-4*x^2)) ) / (2*(1-x));
%o v = Vec(gf)
%o /* _Joerg Arndt_, Jan 24 2013 */
%K nonn
%O 0,3
%A _David Garber_, Sep 19 2005
%E More terms from _Emeric Deutsch_, Dec 31 2006
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