A185417 Table of coefficients of a polynomial sequence related to the Springer numbers.

1, 1, 2, 3, 4, 4, 11, 26, 12, 8, 57, 120, 136, 32, 16, 361, 970, 760, 560, 80, 32, 2763, 7052, 8860, 3680, 2000, 192, 64, 24611, 72530, 72884, 58520, 15120, 6496, 448, 128, 250737, 716528, 976464, 538048, 314720, 55552, 19712, 1024, 256

Offset: 1

Author

Peter Bala, Jan 28 2011

Keywords

• nonn
• easy
• tabl

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).

Examples

			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
...
		

Links

• Peter Bala, The zeros of the row polynomials of A185417

Crossrefs

Cf A001586 (1st column and row sums), A104035, A126156, A147309, A185415, A185418, A185419

Programs

Maple

#A185417
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 *)

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.