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
KEYWORD
AUTHOR
Paul Barry, Apr 23 2009
STATUS
approved