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 A155718 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). 0
 2, -1, -1, 9, -12, 9, -47, 32, 32, -47, 385, -420, 280, -420, 385, -3839, 4354, -1460, -1460, 4354, -3839, 46081, -56490, 26684, -11760, 26684, -56490, 46081, -645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119, 10321921, -14026824 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Row sums are: {2, -2, 6, -30, 210, -1890, 20790, -270270, 4054050, -68918850, 1309458150,...}. The Stirling product form is: as even- odd factorization; 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]}] LINKS FORMULA 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). EXAMPLE {2}, {-1, -1}, {9, -12, 9}, {-47, 32, 32, -47}, {385, -420, 280, -420, 385}, {-3839, 4354, -1460, -1460, 4354, -3839}, {46081, -56490, 26684, -11760, 26684, -56490, 46081}, {-645119, 836296, -418936, 92624, 92624, -418936, 836296, -645119}, {10321921, -14026824, 7562120, -2189376, 718368, -2189376, 7562120, -14026824, 10321921}, {-185794559, 262803366, -150102120, 46239920, -7606032, -7606032, 46239920, -150102120, 262803366, -185794559}, {3715891201, -5441863790, 3264920736, -1076561200, 221207888, -57731520, 221207888, -1076561200, 3264920736, -5441863790, 3715891201} MATHEMATICA Clear[p, x, n, b, a, b0]; p[x_, n_] := Product[x - (2*i), {i, 0, Floor[n/2]}]/x; Table[Expand[ CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 20, 2}]; Flatten[%] CROSSREFS Sequence in context: A176417 A119731 A283321 * A256168 A054768 A104251 Adjacent sequences:  A155715 A155716 A155717 * A155719 A155720 A155721 KEYWORD uned,sign AUTHOR Roger L. Bagula, Jan 25 2009 STATUS approved

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Last modified January 18 11:33 EST 2019. Contains 319271 sequences. (Running on oeis4.)