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A128086 A diagonal of the triangle A128084 of coefficients of q in the q-analog of the even double factorials: a(n) = A128084(n,n).

%I A128086 #6 Sep 25 2013 11:25:41
%S A128086 1,1,2,7,24,86,315,1170,4389,16588,63064,240901,923858,3554747,
%T A128086 13716315,53054703,205651975,798645126,3106669575,12102626404,
%U A128086 47210910670,184385864445,720920510115,2821499709615,11052719207369,43333403693711
%N A128086 A diagonal of the triangle A128084 of coefficients of q in the q-analog of the even double factorials: a(n) = A128084(n,n).
%C A128086 See A128082 for a diagonal of the triangle A128080 of coefficients of q in the q-analog of the odd double factorials.
%e A128086 a(n) is the n-th term in the q-analog of even double factorial (2n)!!, in which the coefficients of q (triangle A128084) begin:
%e A128086 (1);
%e A128086 1,(1);
%e A128086 1,2,(2),2,1;
%e A128086 1,3,5,(7),8,8,7,5,3,1;
%e A128086 1,4,9,16,(24),32,39,44,46,44,39,32,24,16,9,4,1;
%e A128086 1,5,14,30,54,(86),125,169,215,259,297,325,340,340,325,297,...;
%e A128086 The terms enclosed in parenthesis are initial terms of this sequence.
%o A128086 (PARI) a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-q^(2*k))/(1-q)),n,q))
%Y A128086 Cf. A000165 ((2n)!!); A128084 (triangle), A128085 (central terms).
%K A128086 nonn
%O A128086 0,3
%A A128086 _Paul D. Hanna_, Feb 14 2007