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A137456 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).

%I A137456 #11 Aug 06 2025 01:03:00
%S A137456 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 A137456 0,600,0,-3200,0,5280,0,-3200,0,512,120,0,-2880,0,19200,0,-47104,0,
%U A137456 47232,0,-18432,0,2048,0,0,11760,0,-117600,0,385728,0,-560000,0,372736,0,-100352,0,8192
%N A137456 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 A137456 Row sums are:
%C A137456 {1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224}
%C A137456 In real quantum mechanical 2 dimensional orthogonal partitions it would be:
%C A137456   p(x,y,n,m)=T(x,n)*H(y,m).
%C A137456 Here I have made x=y and n=m to get a new sort of polynomial with an odd number of vector coefficients.
%C A137456 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 A137456 p(x,n) = T(x,n)*H(x,n).
%e A137456 Triangle begins:
%e A137456   {1},
%e A137456   {0, 0, 2},
%e A137456   {2, 0, -8, 0, 8},
%e A137456   {0, 0, 36, 0, -72, 0, 32},
%e A137456   {12, 0, -144, 0, 496, 0, -512, 0, 128},
%e A137456   {0, 0, 600, 0, -3200, 0, 5280, 0, -3200, 0, 512},
%e A137456   {120, 0, -2880, 0, 19200, 0, -47104, 0, 47232, 0, -18432, 0, 2048},
%e A137456   {0, 0, 11760, 0, -117600, 0, 385728, 0, -560000, 0,372736, 0, -100352, 0, 8192},
%e A137456   ...
%t A137456 a = Table[CoefficientList[ChebyshevT[n, x]*HermiteH[n, x], x], {n, 0, 10}];
%t A137456 Flatten[a]
%Y A137456 Cf. A053120, A060821.
%K A137456 tabf,uned,sign
%O A137456 1,4
%A A137456 _Roger L. Bagula_, Apr 18 2008