%I A145905
%S A145905 1,1,1,1,3,1,1,9,5,1,1,27,25,7,1,1,81,125,49,9,1,1,243,625,343,81,11,1,
%T A145905 1,729,3125,2401,729,121,13,1,1,2187,15625,16807,6561,1331,169,15,1,1,
%U A145905 6561,78125,117649,59049,14641,2197,225,17,1,1,19683,390625,823543
%N A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.
%C A145905 Definition of the Hilbert transform of a triangular array:
%C A145905 For many square arrays in the database the entries in a row are polynomial 
               in the column index, of degree d say and hence the row generating 
               function has the form P(x)/(1-x)^(d+1), where P is some polynomial 
               function. Often the array whose rows are formed from the coefficients 
               of these P polynomials is of independent interest. This suggests 
               the following definition.
%C A145905 Let [L(n,k)]n,k>=0 be a lower triangular array and let R(n,x) := sum 
               {k = 0 .. n} L(n,k)*x^k, denote the n-th row generating polynomial 
               of L. Then we define the Hilbert transform of L, denoted Hilb(L), 
               to be the square array whose n-th row, n >= 0, has the generating 
               function R(n,x)/(1-x)^(n+1).
%C A145905 In this particular case, L is the array A060187, the array of Eulerian 
               numbers of type B, whose row polynomials are the h-polynomials for 
               permutohedra of type B. The Hilbert transform is an infinite Vandermonde 
               matrix V(1,3,5,...).
%C A145905 We illustrate the Hilbert transform with a few examples:
%C A145905 (1) The Delannoy number array A008288 is the Hilbert transform of Pascal's 
               triangle A007318 (view as the array of coefficients of h-polynomials 
               of n-dimensional cross polytopes).
%C A145905 (2) The transpose of the array of nexus numbers A047969 is the Hilbert 
               transform of the triangle of Eulerian numbers A008292 (best viewed 
               in this context as the coefficients of h-polynomials of n-dimensional 
               permutohedra of type A).
%C A145905 (3) The sequence of Eulerian polynomials begins [1, x, x + x^2, x + 4*x^2 
               + x^3, ...]. The coefficients of these polynomials are recorded in 
               triangle A123125, whose Hilbert transform is A004248 read as square 
               array.
%C A145905 (4) A108625, the array of crystal ball sequences for the A_n lattices, 
               is the Hilbert transform of A008459 (viewed as the triangle of coefficients 
               of h-polynomials of n-dimensional associahedra of type B).
%C A145905 (5) A142992, the array of crystal ball sequences for the C_n lattices, 
               is the Hilbert transform of A086645, the array of h-vectors for type 
               C root polytopes.
%C A145905 (6) A108553, the array of crystal ball sequences for the D_n lattices, 
               is the Hilbert transform of A108558, the array of h-vectors for type 
               D root polytopes.
%C A145905 (7) A086764, read as a square array, is the Hilbert transform of the 
               rencontres numbers A008290.
%C A145905 (8) A143409 is the Hilbert transform of triangle A073107.
%H A145905 Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               vol9.html">On a Number Pyramid Related to the Binomial, Deleham, 
               Eulerian, MacMahon and Stirling number triangles</a>, Journal of 
               Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H A145905 S. Parker, <a href="http://people.brandeis.edu/~gessel/homepage/students/
               parkerthesis.pdf">The Combinatorics of Functional Composition and 
               Inversion</a>, Ph.D. Dissertation, Brandeis Univ. (1993) [From Tom 
               Copeland (tcjpn(AT)msn.com), Nov 09 2008]
%F A145905 T(n,k) = (2*k + 1)^n, (see equation 4.10 in [Franssens]). This array 
               is the infinite Vandermonde matrix V(1,3,5,7, ....) having a LDU 
               factorisation equal to A039755 * diag(2^n*n!) * transpose(A007318).
%e A145905 Triangle A060187 (with an offset of 0) begins
%e A145905 1;
%e A145905 1, 1;
%e A145905 1, 6, 1;
%e A145905 so the entries in the first three rows of the Hilbert transform of
%e A145905 A060187 come from the expansions:
%e A145905 Row 0: 1/(1-x) = 1 + x + x^2 + x^3 + ...;
%e A145905 Row 1: (1+x)/(1-x)^2 = 1 + 3*x + 5*x^2 + 7*x^3 + ...;
%e A145905 Row 2: (1+6*x+x^2)/(1-x)^3 = 1 + 9*x + 25*x^2 + 49*x^3 + ...;
%e A145905 The array begins
%e A145905 n\k|..0....1.....2.....3......4
%e A145905 ================================
%e A145905 0..|..1....1.....1.....1......1
%e A145905 1..|..1....3.....5.....7......9
%e A145905 2..|..1....9....25....49.....81
%e A145905 3..|..1...27...125...343....729
%e A145905 4..|..1...81...625..2401...6561
%e A145905 5..|..1..243..3125.16807..59049
%e A145905 ...
%p A145905 T:=(n,k) -> (2*k + 1)^n: seq(seq(T(n-k,k),k = 0..n),n = 0..10);
%Y A145905 A008292, A039755, A052750 (first superdiagonal), A060187, A114172, A145901.
%Y A145905 Sequence in context: A124496 A074881 A142992 this_sequence A144183 A050153 
               A106340
%Y A145905 Adjacent sequences: A145902 A145903 A145904 this_sequence A145906 A145907 
               A145908
%K A145905 easy,nonn,tabl
%O A145905 0,5
%A A145905 Peter Bala (pbala(AT)toucansurf.com), Oct 27 2008