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A081658 Triangle read by rows, coefficients of polynomials related to the Euler numbers ordered by falling powers. 3
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, 1, 0, -21, 0, 175, 0, -427, 0, 1, 0, -28, 0, 350, 0, -1708, 0, 1385, 1, 0, -36, 0, 630, 0, -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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

These are the coefficients of the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 21 2012

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

LINKS

Table of n, a(n) for n=0..84.

FORMULA

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

From Peter Luschny, Jul 20 2012: (Start)

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

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

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

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

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

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

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

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

EXAMPLE

The triangle begins

[0] 1,

[1] 1, 0,

[2] 1, 0, -1,

[3] 1, 0, -3, 0,

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

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

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

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

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

PROG

(Sage)

R = PolynomialRing(ZZ, 'x')

@CachedFunction

def p(n, x) :

    if n == 0 : return 1

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

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

for n in (0..8) : print A081658_row(n) # Peter Luschny, Jul 20 2012

CROSSREFS

Row reversed: A119879.

Cf. A000364.

Sequence in context: A141665 A136689 A073278 * A187253 A022904 A238341

Adjacent sequences:  A081655 A081656 A081657 * A081659 A081660 A081661

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, Mar 26 2003

EXTENSIONS

Typo in data corrected by Peter Luschny, Jul 20 2012.

STATUS

approved

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)