A137449 A triangular sequence based on concepts of operations on existing sequences: in this case the H(x,n) ( A060821) traditional Hermite is differentiated twice : p(x,n)=-x^2*H''(x,n)+H(x,n).

1, 1, 1, -2, 0, -4, 0, -12, 0, -40, 12, 0, 48, 0, -176, 0, 120, 0, 800, 0, -608, -120, 0, -720, 0, 5280, 0, -1856, 0, -1680, 0, -16800, 0, 25536, 0, -5248, 1680, 0, 13440, 0, -147840, 0, 103936, 0, -14080, 0, 30240, 0, 403200, 0, -919296, 0, 377856, 0, -36352, -30240, 0, -302400, 0, 4435200, 0, -4677120, 0

Offset

1,4

Comments

Row sums are:

{1, 2, -6, -52, -116, 312, 2584, 1808, -42864, -144352, 601504};

As an operator algebra like an Energy Hamiltonian:

e(n)*H(x,n)=p(x,n)/x^2

The relative energy of the row sums goes up much faster than in the Chebyshev

of the first kind.

Links

Table of n, a(n) for n=1..63.

Formula

p(x,n)=-x^2*H''(x,n)+H(x,n)

Example

{1},
{1, 1},
{-2, 0, -4},
{0, -12, 0, -40},
{12, 0, 48, 0, -176},
{0, 120,0, 800, 0, -608},
{-120, 0, -720, 0, 5280, 0, -1856},
{0, -1680, 0, -16800, 0, 25536, 0, -5248},
{1680, 0, 13440, 0, -147840, 0, 103936, 0, -14080},
{0, 30240, 0, 403200, 0, -919296, 0, 377856, 0, -36352},
{-30240, 0, -302400, 0, 4435200, 0, -4677120,0, 1267200, 0, -91136}

Mathematica

Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[HermiteH[n, x], {x, 2}] + HermiteH[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A066493 A278510 A285773 * A056946 A225740 A349479

Adjacent sequences: A137446 A137447 A137448 * A137450 A137451 A137452

Keyword

tabl,uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

Status

approved