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

%I A185417 #13 Apr 15 2015 11:00:27
%S A185417 1,1,2,3,4,4,11,26,12,8,57,120,136,32,16,361,970,760,560,80,32,2763,
%T A185417 7052,8860,3680,2000,192,64,24611,72530,72884,58520,15120,6496,448,
%U A185417 128,250737,716528,976464,538048,314720,55552,19712,1024,256
%N A185417 Table of coefficients of a polynomial sequence related to the Springer numbers.
%C A185417 Define a polynomial sequence S(n,x) recursively by
%C A185417 (1)... S(n+1,x) = x*S(n,x-1)+(x+1)*S(n,x+1) with S(0,x) = 1.
%C A185417 This table lists the coefficients of these polynomials (for n>=1) in ascending powers of x.
%C A185417 The first few polynomials are
%C A185417 S(0,x) = 1
%C A185417 S(1,x) = 2*x+1
%C A185417 S(2,x) = 4*x^2+4*x+3
%C A185417 S(3,x) = 8*x^3+12*x^2+26*x+11.
%C A185417 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.
%C A185417 Compare the recurrence (1) with the recurrence relation satisfied by the coefficients T(n,k) of the polynomials of A104035, namely
%C A185417 (2)... T(n+1,k) = k*T(n,k-1)+(k+1)*T(n,k+1).
%H A185417 Peter Bala, <a href="/A185417/a185417.txt">The zeros of the row polynomials of A185417 </a>
%F A185417 E.g.f: F(x,t) = 1/(cos(t)-sin(t))*(tan(2*t)+sec(2*t))^x
%F A185417 = (cos(t)+sin(t))^x/(cos(t)-sin(t))^(x+1)
%F A185417 = 1 + (2*x+1)*t + (4*x^2+4*x+3)*t^2/2! + ....
%F A185417 Note that (tan(t)+sec(t))^x is the e.g.f for table A147309.
%F A185417 ROW POLYNOMIALS
%F A185417 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.
%F A185417 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.
%F A185417 RELATION WITH OTHER SEQUENCES
%F A185417 1st column [1,1,3,11,57,...] is A001586.
%F A185417 Row sums sequence [1,3,11,57,...] is also A001586.
%F A185417 For n>=1, the values 1/2^n*P(2*n,-1/2) = [1,7,139,5473,...] appear to be A126156.
%e A185417 Table begin
%e A185417 n\k|.....0.....1.....2.....3.....4.....5......6
%e A185417 ===============================================
%e A185417 0..|.....1
%e A185417 1..|.....1.....2
%e A185417 2..|.....3.....4.....4
%e A185417 3..|....11....26....12.....8
%e A185417 4..|....57...120...136....32...16
%e A185417 5..|...361...970...760...560...80.....32
%e A185417 6..|..2763..7052..8860..3680..2000...192....64
%e A185417 ...
%p A185417 #A185417
%p A185417 S := proc(n,x) option remember;
%p A185417 description 'polynomials S(n,x)'
%p A185417 if n = 0 return 1 else return x*S(n-1,x-1)+(x+1)*S(n-1,x+1)
%p A185417 end proc:
%p A185417 with(PolynomialTools):
%p A185417 for n from 1 to 10 CoefficientList(S(n,x),x); end do;
%t A185417 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 *)
%Y A185417 Cf A001586 (1st column and row sums), A104035, A126156, A147309, A185415, A185418, A185419
%K A185417 nonn,easy,tabl
%O A185417 1,3
%A A185417 _Peter Bala_, Jan 28 2011