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A136645 Triangle of coefficients of a Pascal sum of recursive orthogonal Hermite polynomials given in Hochstadt's book: P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].

%I A136645 #6 May 28 2013 03:11:01
%S A136645 1,1,1,-1,2,1,-5,-2,3,1,-3,-16,-3,4,1,21,-12,-35,-4,5,1,43,104,-33,
%T A136645 -64,-5,6,1,-97,246,315,-74,-105,-6,7,1,-455,-656,859,752,-145,-160,
%U A136645 -7,8,1,361,-3402,-2565,2340,1551,-258,-231,-8,9,1,4951,3196,-14805,-7608,5445,2892,-427,-320,-9,10,1
%N A136645 Triangle of coefficients of a Pascal sum of recursive orthogonal Hermite polynomials given in Hochstadt's book: P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].
%C A136645 Row sums are:
%C A136645 {1, 2, 2, -3, -17, -24, 52, 287, 197, -2202, -6674};
%C A136645 The idea here is that the Pascal triangle Binomial heights in the limit give a very normal/ Gaussian-like curve, so that these sums would, in the limit of large n as this linear sum, be more Hermite than other linear sums.
%C A136645 The x^0 constants are, first column:
%C A136645 {1, 1, -1, -5, -3, 21, 43, -97, -455, 361, 4951}
%D A136645 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.
%F A136645 P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].
%e A136645 {1},
%e A136645 {1, 1},
%e A136645 {-1, 2, 1},
%e A136645 {-5, -2, 3, 1},
%e A136645 {-3, -16, -3, 4, 1},
%e A136645 {21, -12, -35, -4, 5, 1},
%e A136645 {43, 104, -33, -64, -5, 6, 1},
%e A136645 {-97, 246, 315, -74, -105, -6, 7,1},
%e A136645 {-455, -656, 859, 752, -145, -160, -7, 8, 1},
%e A136645 {361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1},
%e A136645 {4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1}
%t A136645 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] = Sum[Binomial[n, m]*P[x, m], {m, 0, n}]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P2[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A136645 Cf. A137286.
%K A136645 uned,tabl,sign
%O A136645 1,5
%A A136645 _Roger L. Bagula_, Apr 01 2008