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%I #2 Mar 30 2012 17:34:26
%S 1,0,2,2,0,4,0,10,0,12,24,0,36,0,32,0,148,0,140,0,86,432,0,656,0,512,
%T 0,232,0,3076,0,2976,0,1782,0,624,10112,0,15752,0,12688,0,6040,0,1680,
%U 0,80308,0,80104,0,51148,0,19976,0,4512,188320,0,459736,0,382592,0,198688
%N A triangular sequence of eight back recursive polynomials that are Hermite H(x,n) like and alternating orthogonal on domain {-Infinity,Infinity} and weight function Exp[ -x^2/2]:k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}].
%C Row sums are:
%C {1, 2, 6, 22, 92, 374, 1832, 8458, 46272, 236048, 1306268};
%C The alternating orthogonal integration is:
%C Table[Integrate[P[x, n]*P[x, m]*Exp[ -x^2/2], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0,10}] // TableForm;
%C This sequence is the result of a thought experiment for 8th derivatives.
%C The lower 7 row sums are the same as k=6: only
%C the higher values are really different.
%F k=8 P(x, n) = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P(x, n - m), n^(m/2)*P(x, n - m)], {m, 1, k}]; out_n,m=Coefficients(P(x,n)).
%e {1},
%e {0, 2},
%e {2, 0, 4},
%e {0, 10, 0, 12},
%e {24, 0, 36, 0, 32},
%e {0, 148, 0, 140, 0, 86},
%e {432, 0, 656, 0, 512, 0, 232},
%e {0, 3076, 0, 2976, 0, 1782, 0, 624},
%e {10112, 0, 15752, 0, 12688, 0, 6040, 0, 1680},
%e {0, 80308, 0, 80104, 0, 51148, 0, 19976, 0, 4512},
%e {188320, 0, 459736, 0, 382592, 0, 198688, 0, 64800, 0, 12132}
%t Clear[P, x]:k=8; Table[P[x, -n] = 0, {n, 1, k}]; P[x, 0] = 1; P[x_, n_] := P[x, n] = Sum[If[Mod[m, 2] == 1, (m + 1)*x^m*P[x, n - m], n^(m/2)*P[x, n - m]], {m, 1, k}];; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];
%Y Similar to but different from A138093.
%K nonn,uned,tabl
%O 1,3
%A _Roger L. Bagula_, May 02 2008
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