A136645 Triangle of coefficients of a Pascal sum of recursive orthogonal Hermite polynomials given in Hochstadt's book: P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].

1, 1, 1, -1, 2, 1, -5, -2, 3, 1, -3, -16, -3, 4, 1, 21, -12, -35, -4, 5, 1, 43, 104, -33, -64, -5, 6, 1, -97, 246, 315, -74, -105, -6, 7, 1, -455, -656, 859, 752, -145, -160, -7, 8, 1, 361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1, 4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1

Offset: 1

Author

Roger L. Bagula, Apr 01 2008

Keywords

• uned
• tabl
• sign

Comments

Row sums are:

{1, 2, 2, -3, -17, -24, 52, 287, 197, -2202, -6674};

The idea here is that the Pascal triangle Binomial heights in the limit give a very normal/ Gaussian-like curve, so that these sums would, in the limit of large n as this linear sum, be more Hermite than other linear sums.

The x^0 constants are, first column:

{1, 1, -1, -5, -3, 21, 43, -97, -455, 361, 4951}

Examples

			{1},
{1, 1},
{-1, 2, 1},
{-5, -2, 3, 1},
{-3, -16, -3, 4, 1},
{21, -12, -35, -4, 5, 1},
{43, 104, -33, -64, -5, 6, 1},
{-97, 246, 315, -74, -105, -6, 7,1},
{-455, -656, 859, 752, -145, -160, -7, 8, 1},
{361, -3402, -2565, 2340, 1551, -258, -231, -8, 9, 1},
{4951, 3196, -14805, -7608, 5445, 2892, -427, -320, -9, 10, 1}
		

References

• Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, pp. 8, 42-43.

Crossrefs

Cf. A137286.

Programs

Mathematica

P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = Sum[Binomial[n, m]*P[x, m], {m, 0, n}]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P2[x, n], x], {n, 0, 10}]; Flatten[a]

Formula

P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); p2(x,n)=Sum[Binomial[n,m],{m,0,n}].