cp's OEIS Frontend

A119275 Inverse of triangle related to Padé approximation of exp(x).

%I A119275 #37 Feb 16 2025 08:33:01
%S A119275 1,-2,1,0,-6,1,0,12,-12,1,0,0,60,-20,1,0,0,-120,180,-30,1,0,0,0,-840,
%T A119275 420,-42,1,0,0,0,1680,-3360,840,-56,1,0,0,0,0,15120,-10080,1512,-72,1,
%U A119275 0,0,0,0,-30240,75600,-25200,2520,-90,1,0,0,0,0,0,-332640,277200,-55440,3960,-110,1
%N A119275 Inverse of triangle related to Padé approximation of exp(x).
%C A119275 Inverse of A119274.
%C A119275 Row sums are (-1)^(n+1)*A000321(n+1).
%C A119275 Bell polynomials of the second kind B(n,k)(1,-2). - _Vladimir Kruchinin_, Mar 25 2011
%C A119275 Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) giving unsigned values and adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%H A119275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%F A119275 T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
%F A119275 From _Peter Bala_, May 07 2012: (Start)
%F A119275 E.g.f.: exp(x*(t-t^2)) - 1 = x*t + (-2*x+x^2)*t^2/2! + (-6*x^2+x^3)*t^3/3! + (12*x^2-12*x^3+x^4)*t^4/4! + .... Cf. A059344. Let D denote the operator sum {k >= 0} (-1)^k/k!*x^k*(d/dx)^(2*k). The n-th row polynomial R(n,x) = D(x^n) and satisfies the recurrence equation R(n+1,x) = x*R(n,x)-2*n*x*R(n-1,x). The e.g.f. equals D(exp(x*t)).
%F A119275 (End)
%F A119275 From _Tom Copeland_, Oct 11 2016: (Start)
%F A119275 With initial index n = 1 and unsigned, these are the partition row polynomials of A130561 and A231846 with c_1 = c_2 = x and c_n = 0 otherwise. The first nonzero, unsigned element of each diagonal is given by A001813 (for each row, A001815) and dividing along the corresponding diagonal by this element generates A098158 with its first column removed (cf. A034839 and A086645).
%F A119275 The n-th polynomial is generated by (x - 2y d/dx)^n acting on 1 and then evaluated at y = x, e.g., (x - 2y d/dx)^2 1 = (x - 2y d/dx) x = x^2 - 2y evaluated at y = x gives p_2(x) = -2x + x^2.
%F A119275 (End)
%e A119275 Triangle begins
%e A119275 1,
%e A119275 -2, 1,
%e A119275 0, -6, 1,
%e A119275 0, 12, -12, 1,
%e A119275 0, 0, 60, -20, 1,
%e A119275 0, 0, -120, 180, -30, 1,
%e A119275 0, 0, 0, -840, 420, -42, 1,
%e A119275 0, 0, 0, 1680, -3360, 840, -56, 1,
%e A119275 0, 0, 0, 0, 15120, -10080, 1512, -72, 1
%e A119275 Row 4: D(x^4) = (1 - x*(d/dx)^2 + x^2/2!*(d/dx)^4 - ...)(x^4) = x^4 - 12*x^3 + 12*x^2.
%p A119275 # The function BellMatrix is defined in A264428.
%p A119275 # Adds (1,0,0,0, ..) as column 0.
%p A119275 BellMatrix(n -> `if`(n<2,(n+1)*(-1)^n,0), 9); # _Peter Luschny_, Jan 27 2016
%t A119275 Table[(-1)^(n - k) (n - k)!*Binomial[n + 1, k + 1] Binomial[k + 1, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Oct 12 2016 *)
%t A119275 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A119275 rows = 12;
%t A119275 M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
%t A119275 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A119275 (Sage) # uses[inverse_bell_matrix from A265605]
%o A119275 # Unsigned values and an additional first column (1,0,0, ...).
%o A119275 multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
%o A119275 inverse_bell_matrix(multifact_4_2, 9) # _Peter Luschny_, Dec 31 2015
%Y A119275 Cf. A059344 (unsigned row reverse).
%Y A119275 Cf. A034839, A001813, A001815, A086645, A098158, A130561, A231846.
%K A119275 easy,sign,tabl
%O A119275 0,2
%A A119275 _Paul Barry_, May 12 2006