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A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).

%I A055133 #37 Aug 25 2019 03:09:59
%S A055133 1,-1,1,3,-4,1,-19,27,-9,1,211,-304,108,-16,1,-3651,5275,-1900,300,
%T A055133 -25,1,90921,-131436,47475,-7600,675,-36,1,-3081513,4455129,-1610091,
%U A055133 258475,-23275,1323,-49,1,136407699,-197216832,71282064,-11449536,1033900,-59584,2352,-64,1
%N A055133 Matrix inverse of A008459 (squares of entries of Pascal's triangle).
%C A055133 Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - _Peter Bala_, Jul 24 2013
%H A055133 Alois P. Heinz, <a href="/A055133/b055133.txt">Rows n = 0..99, flattened</a>
%H A055133 J. Riordan, <a href="http://www.jstor.org/stable/2312584">Inverse relations and combinatorial identities</a>, Amer. Math. Monthly, 71 (1964), 485-498; see p. 493 with beta_{n,k} = |T(n,k)|.
%H A055133 W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, 308(24) (2008), 6466-6500.
%F A055133 T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.
%F A055133 From _Peter Bala_, Jul 24 2013: (Start)
%F A055133 Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0,2*sqrt(-y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y + (3 - 4*x + x^2)*y^2/2!^2 + (-19 + 27*x - 9*x^2 + x^3)*y^3/3!^2 + ....
%F A055133 The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A008459 has a generating function E(y)*E(x*y).
%F A055133 Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} binomial(n,k)^2*R(k,x) with initial value R(0,x) = 1.
%F A055133 There appears to be a connection between the zeros of the Bessel function E(x) and the real zeros of the row polynomials R(n,x). Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -1.44579 64907 ... ( = -(A115365/2)^2). It appears that the real zeros of R(n,x) approach zeros of E(alpha*x) as n increases. A numerical example is given below. Indeed, it may be the case that lim_{n -> inf} R(n,x)/R(n,0) = E(alpha*x) for arbitrary complex x. (End)
%e A055133 Table T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A055133       1;
%e A055133      -1,       1;
%e A055133       3,      -4,     1;
%e A055133     -19,      27,    -9,     1;
%e A055133     211,    -304,   108,   -16,   1;
%e A055133   -3651,    5275, -1900,   300, -25,   1;
%e A055133   90921, -131436, 47475, -7600, 675, -36, 1;
%e A055133   ... [edited by _Petros Hadjicostas_, Aug 24 2019]
%e A055133 From _Peter Bala_, Jul 24 2013: (Start)
%e A055133 Function   |        Real zeros to 5 decimal places
%e A055133 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A055133 R(5,x)     | 1, 5.40649,  7.23983
%e A055133 R(10,x)    | 1, 5.26894, 12.97405, 18.53109
%e A055133 R(15,x)    | 1, 5.26894, 12.94909, 24.04769, 33.87883
%e A055133 R(20,x)    | 1, 5.26894, 12.94909, 24.04216, 38.54959, 53.32419
%e A055133 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A055133 E(alpha*x) | 1, 5.26894, 12.94909, 24.04216, 38.54835, 56.46772, ...
%e A055133 where alpha = -1.44579 64907 ... ( = -(A115365/2)^2).
%e A055133 Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( (BesselJZeros(0,n)/BesselJZeros(0,1))^2 ). (End)
%p A055133 T:= proc(n) local M;
%p A055133        M:= Matrix(n+1, (i, j)-> binomial(i-1, j-1)^2)^(-1);
%p A055133        seq(M[n+1, i], i=1..n+1)
%p A055133     end:
%p A055133 seq(T(n), n=0..10);  # _Alois P. Heinz_, Mar 14 2013
%t A055133 T[n_] := Module[{M}, M = Table[Binomial[i-1, j-1]^2, {i, 1, n+1}, {j, 1, n+1}] // Inverse; Table[M[[n+1, i]], {i, 1, n+1}]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Nov 28 2015, after _Alois P. Heinz_ *)
%Y A055133 Cf. A000275, A008459 (matrix inverse), A115365.
%K A055133 sign,tabl
%O A055133 0,4
%A A055133 _Christian G. Bower_, Apr 25 2000