A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1

Offset: 0

Author

Paul D. Hanna, Mar 14 2005

Keywords

• sign
• tabl

Comments

Exponential Riordan array [1 - x, x - x^2/2]; cf. A049403. - Peter Bala, Apr 08 2013

Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Rows begin:
   1;
  -1,  1;
   0, -3,   1;
   0,  3,  -6,    1;
   0,  0,  15,  -10,    1;
   0,  0, -15,   45,  -15,     1;
   0,  0,   0, -105,  105,   -21,     1;
   0,  0,   0,  105, -420,   210,   -28,   1;
   0,  0,   0,    0,  945, -1260,   378, -36,   1;
   0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497:
    1;
    1,   1;
    3,   3,   1;
   15,  15,   6,   1;
  105, 105,  45,  10,  1;
  945, 945, 420, 105, 15, 1; ...
		

Links

• G. C. Greubel, Rows n=0..35 of triangle, flattened
• H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]

Crossrefs

Row sums are found in A001464 (offset 1).

Absolute row sums equal A000085.

Cf. A001497, A049403, A096713, A122848, A130757.

Programs

Mathematica

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

Sage

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016

Formula

E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013