This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A159834 #21 Aug 19 2018 13:43:51
%S A159834 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,
%T A159834 -6,1,20,112,-126,-70,70,0,-7,1,-132,160,448,-336,-140,112,0,-8,1,-28,
%U A159834 -1188,720,1344,-756,-252,168,0,-9,1
%N A159834 Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
%C A159834 Exponential Riordan array [exp(-x-x^2/2), x].
%H A159834 G. C. Greubel, <a href="/A159834/b159834.txt">Rows n=0..100 of triangle, flattened</a>
%F A159834 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).
%F A159834 From _Tom Copeland_, Jun 26 2018: (Start)
%F A159834 E.g.f.: exp[t*p.(x)] = exp[-(t + t^2/2)] e^(x*t).
%F A159834 T(n,k) = binomial(n,k) * A001464(n-k).
%F A159834 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.
%F A159834 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.
%F A159834 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.
%F A159834 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)
%e A159834 Triangle begins:
%e A159834 1,
%e A159834 -1, 1,
%e A159834 0, -2, 1,
%e A159834 2, 0, -3, 1,
%e A159834 -2, 8, 0, -4, 1,
%e A159834 -6, -10, 20, 0, -5, 1,
%e A159834 16, -36, -30, 40, 0, -6, 1,
%e A159834 20, 112, -126, -70, 70, 0, -7, 1,
%e A159834 -132, 160, 448, -336, -140, 112, 0, -8, 1
%e A159834 Production matrix is:
%e A159834 -1, 1,
%e A159834 -1, -1, 1,
%e A159834 0, -2, -1, 1,
%e A159834 0, 0, -3, -1, 1,
%e A159834 0, 0, 0, -4, -1, 1,
%e A159834 0, 0, 0, 0, -5, -1, 1,
%e A159834 0, 0, 0, 0, 0, -6, -1, 1,
%e A159834 0, 0, 0, 0, 0, 0, -7, -1, 1
%p A159834 Trow := proc(n) local b, f; b := proc(n, m) option remember; if n < m or m < 0 then
%p A159834 0 elif n = 0 and m = 0 then 1 else b(n-1, m) + b(n-1, m-1) fi end:
%p A159834 f := proc(n) option remember; if n = 0 then 1 elif n = 1 then -1
%p A159834 else f(n-2) - f(n-1) - f(n-2)*n fi end; seq(b(n, k)*f(n-k), k=0..n) end:
%p A159834 seq(Trow(n), n=0..20); # _Peter Luschny_, Aug 19 2018
%t A159834 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 *)
%o A159834 (PARI) row(n) = apply(x->round(x), Vecrev(polhermite(n, (x-1)/sqrt(2))/ (sqrt(2))^n));
%o A159834 tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Aug 11 2018
%Y A159834 Inverse of A111062.
%Y A159834 Equal to A066325*(A007318)^{-1}.
%Y A159834 First column is A001464.
%Y A159834 Row sums are (-1)^n*A001147(n) aerated.
%Y A159834 Cf. A133314.
%K A159834 easy,sign,tabl
%O A159834 0,5
%A A159834 _Paul Barry_, Apr 23 2009