%I #3 Mar 30 2012 17:34:23
%S 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 246,0,171,0,26,0,1,336,0,888,0,325,0,34,0,1,0,2616,0,2455,0,561,
%U 0,43,0,1,3456,0,11670,0,5745,0,903,0,53,0,1
%N 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,n2).
%C Row sums are:
%C {0, 1, 0, 3, 1, 16, 20, 100, 260, 680, 3320}
%C The double function Integration is alternating:
%C Table[Integrate[Exp[ x^2/2]*P2[x, n]*P2[x, m], {x, Infinity, Infinity}], {n, 0, 10}, {m, 0, 10}];
%C Four Initial conditions were necessary for starting this recursion:
%C P[x, 0] = 1; P[x, 1] = x; P[x, 1] = 0; P[x, 2] = 1;
%F H2(x,n)=A137286(x,n)+A137286(x,n2)
%e {0},
%e {0, 1},
%e {1, 0, 1},
%e {0, 4, 0, 1},
%e {6, 0, 8, 0, 1},
%e {0, 28, 0, 13, 0, 1},
%e {40, 0, 78, 0, 19, 0, 1},
%e {0, 246, 0, 171, 0, 26, 0,1},
%e {336, 0, 888, 0, 325, 0, 34, 0, 1},
%e {0, 2616, 0, 2455, 0, 561, 0, 43, 0, 1},
%e {3456, 0, 11670, 0, 5745, 0, 903, 0, 53, 0, 1}
%t 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 Cf. A137286.
%K uned,tabl,sign
%O 1,8
%A _Roger L. Bagula_, Mar 30 2008
