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%I #12 Feb 04 2022 06:56:56
%S 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,
%T 1,1,729,3125,2401,729,121,13,1,1,2187,15625,16807,6561,1331,169,15,1,
%U 1,6561,78125,117649,59049,14641,2197,225,17,1,1,19683,390625,823543
%N Square array read by antidiagonals: Hilbert transform of triangle A060187.
%C Definition of the Hilbert transform of a triangular array:
%C 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 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 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 We illustrate the Hilbert transform with a few examples:
%C (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 (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 (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 (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 (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 (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 (7) A086764, read as a square array, is the Hilbert transform of the rencontres numbers A008290.
%C (8) A143409 is the Hilbert transform of triangle A073107.
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.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 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_, Nov 09 2008]
%F 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 factorization equal to A039755 * diag(2^n*n!) * transpose(A007318).
%e Triangle A060187 (with an offset of 0) begins
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e so the entries in the first three rows of the Hilbert transform of
%e A060187 come from the expansions:
%e Row 0: 1/(1-x) = 1 + x + x^2 + x^3 + ...;
%e Row 1: (1+x)/(1-x)^2 = 1 + 3*x + 5*x^2 + 7*x^3 + ...;
%e Row 2: (1+6*x+x^2)/(1-x)^3 = 1 + 9*x + 25*x^2 + 49*x^3 + ...;
%e The array begins
%e n\k|..0....1.....2.....3......4
%e ================================
%e 0..|..1....1.....1.....1......1
%e 1..|..1....3.....5.....7......9
%e 2..|..1....9....25....49.....81
%e 3..|..1...27...125...343....729
%e 4..|..1...81...625..2401...6561
%e 5..|..1..243..3125.16807..59049
%e ...
%p T:=(n,k) -> (2*k + 1)^n: seq(seq(T(n-k,k),k = 0..n),n = 0..10);
%Y Cf. A008292, A039755, A052750 (first superdiagonal), A060187, A114172, A145901.
%K easy,nonn,tabl
%O 0,5
%A _Peter Bala_, Oct 27 2008
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