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%I #3 Mar 30 2012 17:34:33
%S 2,-1,-1,9,-12,9,-47,32,32,-47,385,-420,280,-420,385,-3839,4354,-1460,
%T -1460,4354,-3839,46081,-56490,26684,-11760,26684,-56490,46081,
%U -645119,836296,-418936,92624,92624,-418936,836296,-645119,10321921,-14026824
%N Symmetrical form of A039683 using polynomials: p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x; t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n)); t(n,m)=A039683(n,m)+A039683(n,n-m).
%C Row sums are:
%C {2, -2, 6, -30, 210, -1890, 20790, -270270, 4054050, -68918850, 1309458150,...}.
%C The Stirling product form is: as even- odd factorization;
%C Product[x-i,{i,0,n}]=Product[x-(2*i),{i,0,Floor[n/2]}]*Product[x-(2*i+1),{i,0,Floor[n/2]}]
%F p(x,n)=Product[x - (2*i), {i, 0, Floor[n/2]}]/x;
%F t(n,m)=coefficients(p(x,n)+x^n*p(1/x,n));
%F t(n,m)=A039683(n,m)+A039683(n,n-m).
%e {2},
%e {-1, -1},
%e {9, -12, 9},
%e {-47, 32, 32, -47},
%e {385, -420, 280, -420, 385},
%e {-3839, 4354, -1460, -1460, 4354, -3839},
%e {46081, -56490, 26684, -11760, 26684, -56490, 46081},
%e {-645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119},
%e {10321921, -14026824, 7562120, -2189376, 718368, -2189376, 7562120, -14026824, 10321921},
%e {-185794559, 262803366, -150102120, 46239920, -7606032, -7606032, 46239920, -150102120, 262803366, -185794559},
%e {3715891201, -5441863790, 3264920736, -1076561200, 221207888, -57731520, 221207888, -1076561200, 3264920736, -5441863790, 3715891201}
%t Clear[p, x, n, b, a, b0];
%t p[x_, n_] := Product[x - (2*i), {i, 0, Floor[n/2]}]/x;
%t Table[Expand[ CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 20, 2}];
%t Flatten[%]
%Y A039683, A039757
%K uned,sign
%O 0,1
%A _Roger L. Bagula_, Jan 25 2009
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