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 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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 A099174 A066325 * A178117 A095710 A216416 Adjacent sequences:  A137294 A137295 A137296 * A137298 A137299 A137300 KEYWORD uned,tabl,sign AUTHOR Roger L. Bagula, Mar 14 2008 STATUS approved

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Last modified September 17 22:38 EDT 2021. Contains 347489 sequences. (Running on oeis4.)