0,5

Refer to A145905 for the definition of the Hilbert transform of a lower triangular array. For the Hilbert transform of A008459, the array of type B Narayana numbers, see A108625.

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 06 2009: (Start)

The g.f. of the corresponding number triangle is:

1/(1-x-xy-x^2y/(1-x-x^2y/(1-x-xy-x^2y/(1-x-x^2y/(1-x-xy-x^2y.... (continued fraction). (End)

Row n generating function: 1/(n+1) * 1/(1-x) * Jacobi_P(n,1,1,(1+x)/(1-x)) = N_n(x)/(1-x)^n where N_n(x) denotes the shifted Narayana polynomial N_n(x) = sum{k = 1..n} A001263(k)*x^(k-1) of degree n-1.

Conjectural column n generating function: N_n(x^2)/(1-x)^(2n+1).

The entries in row n are given by the values of a polynomial function p_n(x) at x = 0,1,2,... . The first few are p_1(x) = 2x + 1, p_2(x) = (5x^2 + 5x + 2)/2, p_3(x) = (2x + 1)*(7x^2 + 7x + 6)/6 and p_4(x) = (7x^4 + 14x^3 + 21x^2 + 14x + 4)/4. These polynomials appear to have their zeros on the line Re x = -1/2; that is, the polynomials p_n(-x) appear to satisfy a Riemann hypothesis. The corresponding result for A108625 is true (see A142995 for details).

The array begins

n\k|..0.....1.....2.....3.....4.....5

=====================================

0..|..1.....1.....1.....1.....1.....1

1..|..1.....3.....5.....7.....9....11

2..|..1.....6....16....31....51....76

3..|..1....10....40...105...219...396

4..|..1....15....85...295...771..1681

5..|..1....21...161...721..2331..6083

...

Row 2: (1 + 3x + x^2)/(1 - x)^3 = 1 + 6x + 16x^2 + 31x^3 + ... .

Row 3: (1 + 6x + 6x^2 + x^3)/(1 - x)^4 = 1 + 10x + 40x^2 + 105x^3 + ... .

A001263, A005891 (row 2), A063490 (row 3), A108625 (Hilbert transform of h-vectors of type B associahedra).

Sequence in context: A123970 A129818 A055898 this_sequence A159572 A035582 A156594

Adjacent sequences: A145901 A145902 A145903 this_sequence A145905 A145906 A145907

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Oct 31 2008