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A155719 Triangle t(n,m)=A039757(n,m)+A039757(n,n-m) read by rows.

%I A155719 #6 Mar 30 2012 17:34:33
%S A155719 2,0,0,4,-8,4,-14,14,14,-14,106,-192,172,-192,106,-944,1664,-720,-720,
%T A155719 1664,-944,10396,-19560,12644,-6960,12644,-19560,10396,-135134,264158,
%U A155719 -176358,47334,47334,-176358,264158,-135134,2027026,-4098304,2925880
%N A155719 Triangle t(n,m)=A039757(n,m)+A039757(n,n-m) read by rows.
%C A155719 Row sums are zero if n>0.
%C A155719 Building the symmetric form A(n,m)+A(n,n-m) as here is equivalent to tabulation of the coefficients of a polynomial p_n(x) of order n associated with A(.,.) plus its reverse: t(n,m) = [x^m] ( p_n(x)+x^n*p_n(1/x)), here with p_n(x)=product(x-(2i-1)). Note that the product of the polynomial p_n(x) = sum_{m>=0} A(n,m)*x^m and the polynomial p'_n(x)= sum_{m>=0} A(n,n-m)*x^m is given in terms of (terminating) generating function by a convolution, related to the reversal of the sense of the second index in the A(n,m). So the fact that one can obtain A(n,n-m) by using the reverse polynomial p'_n(x) = x^n/p_n(x) is by no means special to this sequence here. The consequence that A(n,m)+A(n,n-m) defines a left-right symmetric row is then obvious.
%e A155719 2;
%e A155719 0, 0;
%e A155719 4, -8, 4;
%e A155719 -14, 14, 14, -14;
%e A155719 106, -192, 172, -192, 106;
%e A155719 -944, 1664, -720, -720, 1664, -944;
%e A155719 10396, -19560, 12644, -6960, 12644, -19560, 10396;
%e A155719 -135134, 264158, -176358, 47334, 47334, -176358, 264158, -135134;
%e A155719 2027026, -4098304, 2925880, -1062656, 416108, -1062656, 2925880, -4098304, 2027026;
%e A155719 -34459424, 71697024, -53806368, 20516768, -3948000, -3948000, 20516768, -53806368, 71697024, -34459424;
%t A155719 Clear[p, x, n, b, a, b0];
%t A155719 p[x_, n_] := Product[x - (2*i + 1), {i, 0, Floor[n/2]}];
%t A155719 Table[Expand[ CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 20, 2}];
%t A155719 Flatten[%]
%Y A155719 Cf. A039683, A039757
%K A155719 sign,tabl
%O A155719 0,1
%A A155719 _Roger L. Bagula_, Jan 25 2009