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URL: https://oeis.org/A104556

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A104556
Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.
10
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
0,5
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
LINKS
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]
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
EXAMPLE
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; ...
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 *)
PROG
(SageMath) # 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
CROSSREFS
Row sums are found in A001464 (offset 1).
Absolute row sums equal A000085.
Sequence in context: A244483 A292727 A049403 * A116089 A122016 A067882
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Mar 14 2005
STATUS
approved