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

1, -1, 2, -3, -4, 4, 11, -18, -12, 8, 57, 88, -72, -32, 16, -361, 570, 440, -240, -80, 32, -2763, -4332, 3420, 1760, -720, -192, 64, 24611, -38682, -30324, 15960, 6160, -2016, -448, 128, 250737, 393776, -309456, -161728, 63840, 19712, -5376, -1024, 256

Offset: 0

Author

Peter Luschny, Jul 30 2012

Keywords

• sign
• tabl

Comments

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)).

Examples

			[0]     1,
[1]    -1,      2,
[2]    -3,     -4,      4,
[3]    11,    -18,    -12,     8,
[4]    57,     88,    -72,   -32,   16,
[5]  -361,    570,    440,  -240,  -80,    32,
[6] -2763,  -4332,   3420,  1760, -720,  -192,   64,
[7] 24611, -38682, -30324, 15960, 6160, -2016, -448, 128.
		

Programs

Sage

@CachedFunction
def SpringerPoly(n,x) :
    if n == 0 : return 1
    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])
R = PolynomialRing(ZZ, 'x')
def A214554_row(n) : return R(SpringerPoly(n,x)).coeffs()
for n in (0..7) : A214554_row(n)

Formula

p{0}(x) = 1 and for n>0

p{n}(x) = Sum_{0<=k

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