OFFSET
0,2
COMMENTS
Inverse of A119274.
Row sums are (-1)^(n+1)*A000321(n+1).
Bell polynomials of the second kind B(n,k)(1,-2). - Vladimir Kruchinin, Mar 25 2011
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
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial
FORMULA
T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
From Peter Bala, May 07 2012: (Start)
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)).
(End)
From Tom Copeland, Oct 11 2016: (Start)
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).
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.
(End)
EXAMPLE
Triangle begins
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, 420, -42, 1,
0, 0, 0, 1680, -3360, 840, -56, 1,
0, 0, 0, 0, 15120, -10080, 1512, -72, 1
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.
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<2, (n+1)*(-1)^n, 0), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
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 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
PROG
(Sage) # uses[inverse_bell_matrix from A265605]
# Unsigned values and an additional first column (1, 0, 0, ...).
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_4_2, 9) # Peter Luschny, Dec 31 2015
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, May 12 2006
STATUS
approved