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

1, -1, 1, 3, -4, 1, -19, 27, -9, 1, 211, -304, 108, -16, 1, -3651, 5275, -1900, 300, -25, 1, 90921, -131436, 47475, -7600, 675, -36, 1, -3081513, 4455129, -1610091, 258475, -23275, 1323, -49, 1, 136407699, -197216832, 71282064, -11449536, 1033900, -59584, 2352, -64, 1

Offset: 0

Author

Christian G. Bower, Apr 25 2000

Keywords

• sign
• tabl

Comments

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

Examples

			Table T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
      1;
     -1,       1;
      3,      -4,     1;
    -19,      27,    -9,     1;
    211,    -304,   108,   -16,   1;
  -3651,    5275, -1900,   300, -25,   1;
  90921, -131436, 47475, -7600, 675, -36, 1;
  ... [edited by _Petros Hadjicostas_, Aug 24 2019]
From _Peter Bala_, Jul 24 2013: (Start)
Function   |        Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 1, 5.40649,  7.23983
R(10,x)    | 1, 5.26894, 12.97405, 18.53109
R(15,x)    | 1, 5.26894, 12.94909, 24.04769, 33.87883
R(20,x)    | 1, 5.26894, 12.94909, 24.04216, 38.54959, 53.32419
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 5.26894, 12.94909, 24.04216, 38.54835, 56.46772, ...
where alpha = -1.44579 64907 ... ( = -(A115365/2)^2).
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)
		

Links

• Alois P. Heinz, Rows n = 0..99, flattened
• J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498; see p. 493 with beta_{n,k} = |T(n,k)|.
• W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, 308(24) (2008), 6466-6500.

Crossrefs

Cf. A000275, A008459 (matrix inverse), A115365.

Programs

Maple

T:= proc(n) local M;
       M:= Matrix(n+1, (i, j)-> binomial(i-1, j-1)^2)^(-1);
       seq(M[n+1, i], i=1..n+1)
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 14 2013

Mathematica

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 *)

Formula

T(n, k) = (-1)^(n+k)*A000275(n-k)*C(n, k)^2.

From Peter Bala, Jul 24 2013: (Start)

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 + ....

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).

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.

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)