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A159834 Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n. 3
1, -1, 1, 0, -2, 1, 2, 0, -3, 1, -2, 8, 0, -4, 1, -6, -10, 20, 0, -5, 1, 16, -36, -30, 40, 0, -6, 1, 20, 112, -126, -70, 70, 0, -7, 1, -132, 160, 448, -336, -140, 112, 0, -8, 1, -28, -1188, 720, 1344, -756, -252, 168, 0, -9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Exponential Riordan array [exp(-x-x^2/2), x].

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

G.f.: 1/(1-xy+x+x^2/(1-xy+x+2x^2/(1-xy+x+3x^2/(1-xy+x+4x^2/(1-... (continued fraction).

From Tom Copeland, Jun 26 2018: (Start)

E.g.f.: exp[t*p.(x)] = exp[-(t + t^2/2)] e^(x*t).

T(n,k) = binomial(n,k) * A001464(n-k).

These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x - 1 - D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.

The transpose of the production matrix gives a matrix representation of the raising operator R, with left multiplication of the rows of this entry treated as column vectors.

exp(-(D + D^2/2)) x^n= e^(-D^2/2) (x - 1)^n = He_n(x-1) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A001464(n) and He_n(x), the unitary or normalized Hermite polynomials of A066325.

A111062 with the e.g.f. exp[t + t^2/2] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator  x + 1 + D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

EXAMPLE

Triangle begins:

     1,

    -1,    1,

     0,   -2,    1,

     2,    0,   -3,    1,

    -2,    8,    0,   -4,    1,

    -6,  -10,   20,    0,   -5,    1,

    16,  -36,  -30,   40,    0,   -6,    1,

    20,  112, -126,  -70,   70,    0,   -7,    1,

  -132,  160,  448, -336, -140,  112,    0,   -8,    1

Production matrix is:

  -1,  1,

  -1, -1,  1,

   0, -2, -1,  1,

   0,  0, -3, -1,  1,

   0,  0,  0, -4, -1,  1,

   0,  0,  0,  0, -5, -1,  1,

   0,  0,  0,  0,  0, -6, -1,  1,

   0,  0,  0,  0,  0,  0, -7, -1,  1

MAPLE

Trow := proc(n) local b, f; b := proc(n, m) option remember; if n < m or m < 0 then

0 elif n = 0 and m = 0 then 1 else b(n-1, m) + b(n-1, m-1) fi end:

f := proc(n) option remember; if n = 0 then 1 elif n = 1 then -1

else f(n-2) - f(n-1) - f(n-2)*n fi end; seq(b(n, k)*f(n-k), k=0..n) end:

seq(Trow(n), n=0..20); # Peter Luschny, Aug 19 2018

MATHEMATICA

T[n_] := CoefficientList[Series[HermiteH[n, (x-1)/Sqrt[2]], {x, 0, 50}], x]/ (Sqrt[2])^n; Table[T[n], {n, 0, 20}] // Flatten (* G. C. Greubel, May 19 2018 *)

PROG

(PARI) row(n) = apply(x->round(x), Vecrev(polhermite(n, (x-1)/sqrt(2))/ (sqrt(2))^n));

tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Aug 11 2018

CROSSREFS

Inverse of A111062.

Equal to A066325*(A007318)^{-1}.

First column is A001464.

Row sums are (-1)^n*A001147(n) aerated.

Cf. A133314.

Sequence in context: A102587 A272608 A257460 * A274576 A257081 A271484

Adjacent sequences:  A159831 A159832 A159833 * A159835 A159836 A159837

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Apr 23 2009

STATUS

approved

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Last modified February 22 16:40 EST 2019. Contains 320399 sequences. (Running on oeis4.)