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 A081658 Triangle read by rows, coefficients of polynomials related to the Euler numbers ordered by falling powers. 3

%I

%S 1,1,0,1,0,-1,1,0,-3,0,1,0,-6,0,5,1,0,1,-10,0,25,0,1,0,-15,0,75,0,-61,

%T 1,0,-21,0,175,0,-427,0,1,0,-28,0,350,0,-1708,0,1385,1,0,-36,0,630,0,

%U -5124,0,12465,0,1,0,-45,0,1050,0,-12810,0,62325,0,-50521,1,0,-55,0,1650,0,-28182,0,228525,0,-555731,0,1,0,-66,0,2475,0

%N Triangle read by rows, coefficients of polynomials related to the Euler numbers ordered by falling powers.

%C These are the coefficients of the Swiss-Knife polynomials A153641. - _Peter Luschny_, Jul 21 2012

%C Nonzero diagonals of the triangle are of the form A000364(k)C(n+2k,2k)(-1)^k.

%F Coefficients of the polynomials in k in the binomial transform of the expansion of 2/(exp(kx)+exp(-kx)).

%F From _Peter Luschny_, Jul 20 2012: (Start)

%F p{n}(0) = Signed Euler secant numbers A122045.

%F p{n}(1) = Signed Euler tangent numbers A155585.

%F p{n}(2) has e.g.f. 2*exp(x)/(exp(-2*x)+1) A119880.

%F 2^n*p{n}(1/2) = Signed Springer numbers A188458.

%F 3^n*p{n}(1/3) has e.g.f. 2*exp(4*x)/(exp(6*x)+1)

%F 4^n*p{n}(1/4) has e.g.f. 2*exp(5*x)/(exp(8*x)+1).

%F Row sum: A155585 (cf. A009006). Absolute row sum: A003701.

%F The GCD of the rows without the first column: A155457. (End)

%e The triangle begins

%e [0] 1,

%e [1] 1, 0,

%e [2] 1, 0, -1,

%e [3] 1, 0, -3, 0,

%e [4] 1, 0, -6, 0, 5,

%e [5] 1, 0, -10, 0, 25, 0,

%e [6] 1, 0, -15, 0, 75, 0, -61,

%e [7] 1, 0, -21, 0, 175, 0, -427, 0,

%e [8] 1, 0, -28, 0, 350, 0, -1708, 0, 1385.

%o (Sage)

%o R = PolynomialRing(ZZ, 'x')

%o @CachedFunction

%o def p(n, x) :

%o if n == 0 : return 1

%o return add(p(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])

%o def A081658_row(n) : return [R(p(n,x)).reverse()[i] for i in (0..n)]

%o for n in (0..8) : print A081658_row(n) # _Peter Luschny_, Jul 20 2012

%Y Row reversed: A119879.

%Y Cf. A000364.

%K easy,sign,tabl

%O 0,9

%A _Paul Barry_, Mar 26 2003

%E Typo in data corrected by _Peter Luschny_, Jul 20 2012.

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Last modified March 25 03:50 EDT 2019. Contains 321450 sequences. (Running on oeis4.)