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%I #28 Apr 03 2023 15:13:38
%S 1,-1,1,3,-4,1,-25,36,-12,1,543,-800,288,-32,1,-29281,43440,-16000,
%T 1920,-80,1,3781503,-5621952,2085120,-256000,11520,-192,1,-1138779265,
%U 1694113344,-629658624,77844480,-3584000,64512,-448,1,783702329343,-1166109967360,433693016064,-53730869248,2491023360,-45875200,344064,-1024,1
%N Matrix inverse of A111636.
%C Let Q be the class of labeled directed acyclic graphs (dags) with some designated source nodes. Here, a source node is a node of indegree 0 and some means possibly all or none. |a(n,k)| is the number of dags in Q containing n nodes with exactly k designated source nodes. - _Geoffrey Critzer_, Apr 02 2023
%H Vincenzo Librandi, <a href="/A224069/b224069.txt">Rows n = 0..50, flattened</a>
%H W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500, (2008).
%F T(n,k) = (-1)^(n-k)*A003024(n-k)*A111636(n,k) = (-1)^(n-k)*A003024(n-k)*2^(k*(n-k))*binomial(n,k).
%F Sum_{k = 1..n} k*2^k*T(n,k) = 0 for n >= 1.
%F Let E(x) = Sum_{n >= 0} x^n/(n!*2^binomial(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function for this triangle is E(x*z)/E(z) = 1 + (x - 1)*z + (x^2 - 4*x + 3)*z^2/(2!*2) + (x^3 - 12*x^2 + 36*x - 25)*z^3/(3!*2^3) + ....
%F This triangle is a generalized Riordan array (1/E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.
%F The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x^n - Sum_{k = 0..n-1} binomial(n,k)*2^(k*(n-k))*R(k,x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only positive real zeros of multiplicity 1. Moreover, if r(n,1) < r(n,2) < ... < r(n,n) denotes the zeros of R(n,x) arranged in increasing order then it appears that lim_{n -> oo} r(n,i) exists for each fixed 1 <= i <= n. An example is given below.
%e Triangle begins
%e n\k.|......0......1......2......3......4......5
%e = = = = = = = = = = = = = = = = = = = = = = = =
%e .0..|......1
%e .1..|.....-1......1
%e .2..|......3.....-4......1
%e .3..|....-25.....36....-12......1
%e .4..|....543...-800....288....-32......1
%e .5..|.-29281..43440.-16000...1920....-80......1
%e ...
%e The sequence of zeros of R(10,x) begins 1, 3.280147..., 9.112469..., 23.366923..., 57.084317....
%e The sequence of zeros of R(20,x) begins 1, 3.280163..., 9.112696..., 23.369274..., 57.105379....
%t max = 8; A111636 = Table[ Binomial[n, k]*2^(k*(n - k)), {n, 0, max}, {k, 0, max}]; t = Inverse[ A111636 ]; Table[ t[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 10 2013 *)
%Y Cf. A003024 (first column), A111636 (matrix inverse).
%K sign,easy,tabl
%O 0,4
%A _Peter Bala_, Apr 09 2013
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