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A136586 Triangle of coefficients of even modified recursive orthogonal Hermite polynomials given in Hochstadt's book:P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) ;A137286; P2(x,n)=P(x,n)+P(x,n-2).

%I A136586 #3 Mar 30 2012 17:34:23
%S A136586 0,0,1,-1,0,1,0,-4,0,1,6,0,-8,0,1,0,28,0,-13,0,1,-40,0,78,0,-19,0,1,0,
%T A136586 -246,0,171,0,-26,0,1,336,0,-888,0,325,0,-34,0,1,0,2616,0,-2455,0,561,
%U A136586 0,-43,0,1,-3456,0,11670,0,-5745,0,903,0,-53,0,1
%N A136586 Triangle of coefficients of even modified recursive orthogonal Hermite polynomials given in Hochstadt's book:P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) ;A137286; P2(x,n)=P(x,n)+P(x,n-2).
%C A136586 Row sums are:
%C A136586 {0, 1, 0, -3, -1, 16, 20, -100, -260, 680, 3320}
%C A136586 The double function Integration is alternating:
%C A136586 Table[Integrate[Exp[ -x^2/2]*P2[x, n]*P2[x, m], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0, 10}];
%C A136586 Four Initial conditions were necessary for starting this recursion:
%C A136586 P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1;
%F A136586 H2(x,n)=A137286(x,n)+A137286(x,n-2)
%e A136586 {0},
%e A136586 {0, 1},
%e A136586 {-1, 0, 1},
%e A136586 {0, -4, 0, 1},
%e A136586 {6, 0, -8, 0, 1},
%e A136586 {0, 28, 0, -13, 0, 1},
%e A136586 {-40, 0, 78, 0, -19, 0, 1},
%e A136586 {0, -246, 0, 171, 0, -26, 0,1},
%e A136586 {336, 0, -888, 0, 325, 0, -34, 0, 1},
%e A136586 {0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1},
%e A136586 {-3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1}
%t A136586 P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = P[x, n] + P[x, n - 2]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Join[{0}, Table[CoefficientList[P2[x, n], x], {n, 0, 10}]]; Flatten[a]
%Y A136586 Cf. A137286.
%K A136586 uned,tabl,sign
%O A136586 1,8
%A A136586 _Roger L. Bagula_, Mar 30 2008