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Number of set partitions of [n] avoiding the patterns {1123, 1211}.
1

%I #31 May 31 2017 23:01:22

%S 1,1,2,5,13,33,81,196,470,1126,2699,6487,15633,37788,91589,222572,

%T 542145,1323446,3237074,7932108,19469151,47860083,117819348,290424126,

%U 716772644,1771035921,4380646788,10846386691,26880759090,66678169061,165534924098,411281773379,1022621256416,2544478797575,6335428289930,15784538365081,39350771601502,98158461390807

%N Number of set partitions of [n] avoiding the patterns {1123, 1211}.

%H G. C. Greubel, <a href="/A210496/b210496.txt">Table of n, a(n) for n = 0..1000</a>

%H V. Jelinek, T. Mansour, M. Shattuck, <a href="http://dx.doi.org/10.1016/j.aam.2012.09.002">On multiple pattern avoiding set partitions</a>, Adv. Appl. Math. 50 (2) (2013) 292-326, Theorem 4.27.

%F G.f.: ( (1-x^2)*sqrt((1-x)*(1-x-4*x^2)) -(1-3*x-2*x^2 +14*x^3 -15*x^4 +3*x^5) / (1-x)^2 ) / ( 2*x^2*(1-3*x+x^2) ).

%F a(n) ~ sqrt((5389+1307*sqrt(17))/2)*((1+sqrt(17))/2)^n/(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Jul 30 2013

%F Conjecture: (n+2)*a(n) +2*(-3*n-4)*a(n-1) +4*(2*n+3)*a(n-2) +(13*n-40)*a(n-3) +5*(-7*n+18)*a(n-4) +6*(2*n-1)*a(n-5) +22*(n-7)*a(n-6) +(-19*n+134)*a(n-7) +2*(2*n-15)*a(n-8)=0. - _R. J. Mathar_, Oct 08 2016

%t CoefficientList[Series[((1-x^2)*Sqrt[(1-x)*(1-x-4*x^2)]-(1-3*x-2*x^2 +14*x^3-15*x^4+3*x^5)/(1-x)^2)/(2*x^2*(1-3*x+x^2)), {x, 0, 20}], x]

%o (PARI) x='x+O('x^50); Vec(( (1-x^2)*sqrt((1-x)*(1-x-4*x^2)) -(1-3*x-2*x^2+14*x^3-15*x^4+3*x^5) / (1-x)^2 ) / ( 2*x^2*(1-3*x+x^2) )) \\ _G. C. Greubel_, May 31 2017

%K nonn

%O 0,3

%A _R. J. Mathar_, Jan 25 2013

%E Typo in g.f. corrected by _Vaclav Kotesovec_, Jul 30 2013