A136209 Differentiation of A137286: Triangle of coefficients of differentiation recursive orthogonal Hermite polynomials given in Hochstadt's book : P(x, n) = x*P(x, n - 1) - n*P(x, n - 2).
1, 0, 2, -5, 0, 3, 0, -18, 0, 4, 33, 0, -42, 0, 5, 0, 174, 0, -80, 0, 6, -279, 0, 555, 0, -135, 0, 7, 0, -1950, 0, 1380, 0, -210, 0, 8, 2895, 0, -7920, 0, 2940, 0, -308, 0, 9, 0, 25290, 0, -24360, 0, 5628, 0, -432, 0, 10, -35685, 0, 125055, 0, -62790, 0, 9954, 0, -585, 0, 11
Offset: 1
Examples
{1},
{0, 2},
{-5, 0, 3},
{0, -18, 0, 4},
{33, 0, -42, 0,5},
{0, 174, 0, -80, 0, 6},
{-279, 0, 555, 0, -135, 0, 7},
{0, -1950, 0, 1380, 0, -210, 0, 8},
{2895, 0, -7920, 0, 2940, 0, -308, 0, 9},
{0, 25290,0, -24360, 0, 5628, 0, -432, 0, 10},
{-35685, 0, 125055, 0, -62790, 0,9954, 0, -585, 0, 11}
References
- page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
Crossrefs
Cf. A137286.
Programs
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Mathematica
H[x, 0] = 1; H[x, 1] = x; H[x_, n_] := H[x, n] = x*H[x, n - 1] - n*H[x, n - 2]; L[x_, n_] := D[H[x, n + 1], x]; a0 = Table[ExpandAll[L[x, n]], {n, 0, 10}]; a = Table[CoefficientList[L[x, n], x], {n, 0, 10}]; Flatten[a]
Formula
P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); L(x,n)=dP(x,n+1]/dx
Comments