A104556 - OEIS

%I #30 Mar 27 2020 06:58:10

%S 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,

%T -21,1,0,0,0,105,-420,210,-28,1,0,0,0,0,945,-1260,378,-36,1,0,0,0,0,

%U -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

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

%C Exponential Riordan array [1 - x, x - x^2/2]; cf. A049403. - _Peter Bala_, Apr 08 2013

%C 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

%H G. C. Greubel, <a href="/A104556/b104556.txt">Rows n=0..35 of triangle, flattened</a>

%H H. Han and S. Seo, <a href="http://dx.doi.org/10.1016/j.ejc.2007.12.002">Combinatorial proofs of inverse relations and log-concavity for Bessel numbers</a>, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]

%F 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

%e Rows begin:

%e    1;

%e   -1,  1;

%e    0, -3,   1;

%e    0,  3,  -6,    1;

%e    0,  0,  15,  -10,    1;

%e    0,  0, -15,   45,  -15,     1;

%e    0,  0,   0, -105,  105,   -21,     1;

%e    0,  0,   0,  105, -420,   210,   -28,   1;

%e    0,  0,   0,    0,  945, -1260,   378, -36,   1;

%e    0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1; ...

%e The columns being equal in absolute value to the rows of the matrix inverse A001497:

%e     1;

%e     1,   1;

%e     3,   3,   1;

%e    15,  15,   6,   1;

%e   105, 105,  45,  10,  1;

%e   945, 945, 420, 105, 15, 1; ...

%t 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 *)

%o (Sage) # uses[bell_matrix from A264428]

%o # Adds a column 1,0,0,0, ... at the left side of the triangle.

%o bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # _Peter Luschny_, Jan 19 2016

%Y Row sums are found in A001464 (offset 1).

%Y Absolute row sums equal A000085.

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

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 14 2005