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).

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, 0, 600, 0, -3200, 0, 5280, 0, -3200, 0, 512, 120, 0, -2880, 0, 19200, 0, -47104, 0, 47232, 0, -18432, 0, 2048, 0, 0, 11760, 0, -117600, 0, 385728, 0, -560000, 0, 372736, 0, -100352, 0, 8192

Offset: 1

Author

Roger L. Bagula, Apr 18 2008

Keywords

• tabf
• uned
• sign

Comments

Row sums are:

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

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

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

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

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).

Examples

			Triangle begins:
  {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, 0, 600, 0, -3200, 0, 5280, 0, -3200, 0, 512},
  {120, 0, -2880, 0, 19200, 0, -47104, 0, 47232, 0, -18432, 0, 2048},
  {0, 0, 11760, 0, -117600, 0, 385728, 0, -560000, 0,372736, 0, -100352, 0, 8192},
  ...
		

Crossrefs

Cf. A053120, A060821.

Programs

Mathematica

a = Table[CoefficientList[ChebyshevT[n, x]*HermiteH[n, x], x], {n, 0, 10}];
Flatten[a]

Formula

p(x,n) = T(x,n)*H(x,n).