OFFSET

1,3

COMMENTS

Define a polynomial sequence S(n,x) recursively by

(1)... S(n+1,x) = x*S(n,x-1)+(x+1)*S(n,x+1) with S(0,x) = 1.

This table lists the coefficients of these polynomials (for n>=1) in ascending powers of x.

The first few polynomials are

S(0,x) = 1

S(1,x) = 2*x+1

S(2,x) = 4*x^2+4*x+3

S(3,x) = 8*x^3+12*x^2+26*x+11.

The sequence [1,1,3,11,57,...] of constant terms of the polynomials is the sequence of Springer numbers A001586. The zeros of the polynomials S(n,-x) lie on the vertical line Re x = 1/2 in the complex plane.

Compare the recurrence (1) with the recurrence relation satisfied by the coefficients T(n,k) of the polynomials of A104035, namely

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

LINKS

FORMULA

E.g.f: F(x,t) = 1/(cos(t)-sin(t))*(tan(2*t)+sec(2*t))^x

= (cos(t)+sin(t))^x/(cos(t)-sin(t))^(x+1)

= 1 + (2*x+1)*t + (4*x^2+4*x+3)*t^2/2! + ....

Note that (tan(t)+sec(t))^x is the e.g.f for table A147309.

ROW POLYNOMIALS

The easily checked identity d/dt F(x,t) = x*F(x-1,t)+(x+1)*F(x+1,t) shows that the row generating polynomials of this table are the polynomials S(n,x) described in the Comments section above.

The polynomials S(n,-x) satisfy a Riemann hypothesis: that is, the zeros of S(n,-x) lie on the vertical line Re(x) = 1/2 in the complex plane - see the link.

RELATION WITH OTHER SEQUENCES

1st column [1,1,3,11,57,...] is A001586.

Row sums sequence [1,3,11,57,...] is also A001586.

For n>=1, the values 1/2^n*P(2*n,-1/2) = [1,7,139,5473,...] appear to be A126156.

EXAMPLE

Table begin

n\k|.....0.....1.....2.....3.....4.....5......6

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

0..|.....1

1..|.....1.....2

2..|.....3.....4.....4

3..|....11....26....12.....8

4..|....57...120...136....32...16

5..|...361...970...760...560...80.....32

6..|..2763..7052..8860..3680..2000...192....64

...

MAPLE

S := proc(n, x) option remember;

description 'polynomials S(n, x)'

if n = 0 return 1 else return x*S(n-1, x-1)+(x+1)*S(n-1, x+1)

end proc:

with(PolynomialTools):

for n from 1 to 10 CoefficientList(S(n, x), x); end do;

MATHEMATICA

S[0, _] = 1; S[n_, x_] := S[n, x] = x*S[n-1, x-1] + (x+1)*S[n-1, x+1]; Table[ CoefficientList[S[n, x], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Apr 15 2015 *)

CROSSREFS

KEYWORD

AUTHOR

Peter Bala, Jan 28 2011

STATUS

approved