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A214554
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Triangle read by rows, coefficients of polynomials related to the Springer numbers A001586.
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0
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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
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OFFSET
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0,3
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COMMENTS
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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)).
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LINKS
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FORMULA
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p{0}(x) = 1 and for n>0
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).
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EXAMPLE
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[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.
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PROG
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(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()
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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