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A214554 Triangle read by rows, coefficients of polynomials related to the Springer numbers A001586.

%I A214554 #25 Jul 30 2012 13:02:22
%S A214554 1,-1,2,-3,-4,4,11,-18,-12,8,57,88,-72,-32,16,-361,570,440,-240,-80,
%T A214554 32,-2763,-4332,3420,1760,-720,-192,64,24611,-38682,-30324,15960,6160,
%U A214554 -2016,-448,128,250737,393776,-309456,-161728,63840,19712,-5376,-1024,256
%N A214554 Triangle read by rows, coefficients of polynomials related to the Springer numbers A001586.
%C A214554 The polynomials might be called Springer polynomials because both p{n}(0) and p{n}(1) are signed versions of the Springer numbers. p{n}(0) is the first column of the triangle (A212435 with e.g.f. exp(-x)/cosh(2x)) and p{n}(1) are the row sums (A188458 with e.g.f. exp(x)/cosh(2x)).
%F A214554 p{0}(x) = 1 and for n>0
%F A214554 p{n}(x) = Sum_{0<=k<n; 2|k} 2^(n-k)*p{k}(1/2)*binomial(n,k)*((x-1/2)^(n-k)+(n mod 2)-1).
%e A214554 [0]     1,
%e A214554 [1]    -1,      2,
%e A214554 [2]    -3,     -4,      4,
%e A214554 [3]    11,    -18,    -12,     8,
%e A214554 [4]    57,     88,    -72,   -32,   16,
%e A214554 [5]  -361,    570,    440,  -240,  -80,    32,
%e A214554 [6] -2763,  -4332,   3420,  1760, -720,  -192,   64,
%e A214554 [7] 24611, -38682, -30324, 15960, 6160, -2016, -448, 128.
%o A214554 (Sage)
%o A214554 @CachedFunction
%o A214554 def SpringerPoly(n,x) :
%o A214554     if n == 0 : return 1
%o A214554     return add(2^(n-k)*SpringerPoly(k,1/2)*binomial(n,k)*((x-1/2)^(n-k)+n%2-1) for k in range(n)[::2])
%o A214554 R = PolynomialRing(ZZ, 'x')
%o A214554 def A214554_row(n) : return R(SpringerPoly(n,x)).coeffs()
%o A214554 for n in (0..7) : A214554_row(n)
%K A214554 sign,tabl
%O A214554 0,3
%A A214554 _Peter Luschny_, Jul 30 2012