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%I #19 Aug 11 2014 22:45:50
%S 0,0,0,0,0,2,6,18,47,123,318,830,2182,5792,15504,41828,113626,310564,
%T 853458,2356770,6536372,18199274,50852008,142547548,400763211,
%U 1129760403,3192727784,9043402488,25669818462,73007772788,208023278194,593742784814,1697385471195
%N From the enumeration of involutions avoiding the pattern 4321.
%H Alois P. Heinz, <a href="/A217526/b217526.txt">Table of n, a(n) for n = 0..650</a>
%H Piera Manara and Claudio Perelli Cippo, <a href="http://www.mat.unisi.it/newsito/puma/public_html/22_2/manara_perelli-cippo.pdf">The fine structure of 4321 avoiding involutions and 321 avoiding involutions</a>, PU. M. A. Vol. 22 (2011), 227-238. See page 233.
%F Manara and Perelli Cippo give a g.f.:
%F G.f.: (1 - x - sqrt(1 - 2*x - 3*x^2))/2 - x^2/((1 - x)*(1 - x^2)).
%F Recurrence (for n>5): (n-5)*n*(2*n-7)*a(n) = 2*(n-3)*(2*n^2 - 12*n + 15)*a(n-1) + 2*(4*n^3 - 47*n^2 + 177*n - 215)*a(n-2) - 2*(n-4)*(2*n^2 - 6*n - 5)*a(n-3) - 3*(n-5)*(n-4)*(2*n-5)*a(n-4). - _Vaclav Kotesovec_, Aug 18 2013
%F a(n) ~ 3^n/(2*sqrt(3*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 18 2013
%o (PARI) all_a(m) = { x = y+O(y^(m+1)); P = (1 - x - sqrt(1-2*x-3*x^2))/2 - x^2/((1-x)*(1-x^2)); for (n=0, m, print1(polcoeff(P, n, y), ", "));} \\ _Michel Marcus_, Feb 08 2013
%K nonn
%O 0,6
%A _N. J. A. Sloane_, Oct 13 2012
%E More terms from _Michel Marcus_, Feb 08 2013
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