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A triangular sequence of coefficients of a partition two types polynomials; of Chebyshev of the first kind polynomials ( A053120 ) and Hermite polynomials ( A060821): p(x,n)=T(x,n)*H(x,n).
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%I #6 May 28 2013 03:11:23

%S 1,0,0,2,2,0,-8,0,8,0,0,36,0,-72,0,32,12,0,-144,0,496,0,-512,0,128,0,

%T 0,600,0,-3200,0,5280,0,-3200,0,512,120,0,-2880,0,19200,0,-47104,0,

%U 47232,0,-18432,0,2048,0,0,11760,0,-117600,0,385728,0,-560000,0,372736,0,-100352,0,8192,1680,0,-67200,0,712320,0

%N A triangular sequence of coefficients of a partition two types polynomials; of Chebyshev of the first kind polynomials ( A053120 ) and Hermite polynomials ( A060821): p(x,n)=T(x,n)*H(x,n).

%C Row sums are:

%C {1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224}

%C In real quantum mechanical 2 dimensional orthogonal partitions it would be:

%C p(x,y,n,m)=T(x,n)*H(y,m).

%C Here I have made x=y and n=m to get a new sort of polynomial with an odd number of vector coefficients.

%C The traditional Schoedinger wave mechanics solution of hydrogen is a partition of four (not two dimensions): wave_function=Bessel(r,n)*Legendre(theta,l)*Fourier(phi,m)*Spin(t,s).

%F p(x,n)=T(x,n)*H(x,n)

%e {1},

%e {0, 0, 2},

%e {2, 0, -8, 0, 8},

%e {0, 0, 36,0, -72, 0, 32},

%e {12, 0, -144, 0, 496, 0, -512, 0,128},

%e {0, 0, 600, 0, -3200, 0, 5280, 0, -3200, 0, 512},

%e {120, 0, -2880, 0, 19200, 0, -47104, 0, 47232, 0, -18432, 0, 2048},

%e {0, 0, 11760,0, -117600, 0, 385728, 0, -560000, 0,372736, 0, -100352, 0, 8192}, {1680, 0, -67200,0, 712320, 0, -3014144, 0, 5921024, 0, -5742592, 0, 2678784, 0, -524288, 0, 32768},

%e {0, 0, 272160, 0, -4354560, 0, 23175936, 0, -58143744, 0, 76202496,0, -52555776, 0, 17915904, 0, -2654208, 0, 131072},

%e {30240, 0, -1814400, 0, 27619200, 0, -175150080, 0, 546762240, 0, -919803904, 0, 860825600, 0, -439091200, 0, 113213440, 0, -13107200, 0, 524288}

%t Table[ExpandAll[ChebyshevT[n, x]*HermiteH[n, x]], {n, 0, 10}]; a = Table[CoefficientList[ChebyshevT[n, x]*HermiteH[n, x], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[ChebyshevT[n, x]*HermiteH[n,x], x]], {n, 0, 10}].

%Y Cf. A053120, A060821.

%K tabf,uned,sign

%O 1,4

%A _Roger L. Bagula_, Apr 18 2008