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A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

%I A122850 #45 Sep 01 2022 08:59:30
%S A122850 1,0,1,0,-1,1,0,3,-3,1,0,-15,15,-6,1,0,105,-105,45,-10,1,0,-945,945,
%T A122850 -420,105,-15,1,0,10395,-10395,4725,-1260,210,-21,1,0,-135135,135135,
%U A122850 -62370,17325,-3150,378,-28,1,0,2027025,-2027025,945945,-270270,51975,-6930,630,-36,1
%N A122850 Exponential Riordan array (1, sqrt(1+2x)-1).
%C A122850 Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.
%C A122850 Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%H A122850 P. Bala, <a href="/A048993/a048993.pdf">The white diamond product of power series</a>
%H A122850 Orli Herscovici, <a href="https://arxiv.org/abs/1904.07674">Study of the p,q-deformed Touchard polynomials</a>, arXiv:1904.07674 [math.CO], 2019.
%H A122850 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3
%H A122850 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bessel_polynomials">Bessel polynomials</a>
%H A122850 S. Willerton, <a href="https://arxiv.org/abs/1708.03227">The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials</a>, arXiv:1708.03227v1 [math.MG], 2017.
%F A122850 T(n,k) = (-1)^(n-k)*A132062(n,k). - _Philippe Deléham_, Nov 06 2011
%F A122850 Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - _Peter Bala_, Jun 23 2014
%F A122850 From _Peter Bala_, Jan 09 2018: (Start)
%F A122850 Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.
%F A122850 Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.
%F A122850 The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.
%F A122850 If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)
%F A122850 Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - _Alois P. Heinz_, Aug 31 2022
%e A122850 Triangle begins
%e A122850   1
%e A122850   0 1
%e A122850   0 -1 1
%e A122850   0 3 -3 1
%e A122850   0 -15 15 -6 1
%e A122850   0 105 -105 45 -10 1
%e A122850   0 -945 945 -420 105 -15 1
%e A122850   0 10395 -10395 4725 -1260 210 -21 1
%e A122850   0 -135135 135135 -62370 17325 -3150 378 -28 1
%e A122850   0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1
%e A122850   0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1
%e A122850   ...
%p A122850 # The function BellMatrix is defined in A264428.
%p A122850 BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # _Peter Luschny_, Jan 27 2016
%t A122850 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A122850 rows = 12;
%t A122850 M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows];
%t A122850 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 26 2018, after _Peter Luschny_ *)
%o A122850 (Sage) # uses[bell_matrix from A264428]
%o A122850 bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # _Peter Luschny_, Jan 19 2016
%Y A122850 Cf. A000806, A001497, A008277, A039757, A122848, A132062, A144301.
%K A122850 sign,tabl,easy
%O A122850 0,8
%A A122850 _Paul Barry_, Sep 14 2006
%E A122850 More terms from _Alois P. Heinz_, Aug 31 2022