A137297 Triangle read by rows: coefficients of a Hermite-like set of recursive polynomials that appear by integration to be orthogonal using the substitution on the Hermite recursion of n->f(n) where f(n)=A000045[n] is the Fibonacci sequence.

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 18, 0, -11, 0, 1, -24, 0, 66, 0, -19, 0, 1, 0, -258, 0, 209, 0, -32, 0, 1, 504, 0, -1644, 0, 608, 0, -53, 0, 1, 0, 9276, 0, -8750, 0, 1696, 0, -87, 0, 1, -27720, 0, 99696, 0, -42190, 0, 4611, 0, -142, 0, 1

Offset

1,8

Comments

Note that Table[Integrate[Exp[ -x^2/2]*P[x,n]*P[x, n + 1], {x, -Infinity, Infinity}], {n, 0, 10}]; gives {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}.

The row sums are: {1, 1, 0, -2, -2, 8, 24, -80, -584, 2136, 34256, ...}

Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];

Links

Table of n, a(n) for n=1..66.

Formula

a(n) = a(n-1)+a(n-2): A000045(n); p(x,0)=1;p(x,1)=x; p(x,n)=x*p(x,n-1)-a(n)*p(n,n-2)

Example

{1},
{0, 1},
{-1, 0, 1},
{0, -3, 0, 1},
{3, 0, -6, 0, 1},
{0, 18, 0, -11, 0, 1},
{-24, 0, 66, 0, -19, 0, 1},
{0, -258, 0, 209, 0, -32, 0, 1},
{504, 0, -1644, 0, 608, 0, -53, 0, 1},
{0, 9276, 0, -8750, 0, 1696, 0, -87, 0, 1},
{-27720, 0, 99696, 0, -42190, 0, 4611, 0, -142, 0, 1}

Mathematica

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

Crossrefs

Cf. A000045.

Sequence in context: A179898 A066325 A099174 * A178117 A095710 A216416

Adjacent sequences: A137294 A137295 A137296 * A137298 A137299 A137300

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Mar 14 2008

Status

approved