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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).
0
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, 1680, 0, -67200, 0, 712320, 0
OFFSET
1,4
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).
FORMULA
p(x,n)=T(x,n)*H(x,n)
EXAMPLE
{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}, {1680, 0, -67200,0, 712320, 0, -3014144, 0, 5921024, 0, -5742592, 0, 2678784, 0, -524288, 0, 32768},
{0, 0, 272160, 0, -4354560, 0, 23175936, 0, -58143744, 0, 76202496,0, -52555776, 0, 17915904, 0, -2654208, 0, 131072},
{30240, 0, -1814400, 0, 27619200, 0, -175150080, 0, 546762240, 0, -919803904, 0, 860825600, 0, -439091200, 0, 113213440, 0, -13107200, 0, 524288}
MATHEMATICA
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}].
CROSSREFS
Sequence in context: A060007 A021457 A305605 * A337710 A248948 A364288
KEYWORD
tabf,uned,sign
AUTHOR
Roger L. Bagula, Apr 18 2008
STATUS
approved