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A119879 Exponential Riordan array (sech(x),x). 13
1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1, 0, 12465, 0, -5124, 0, 630, 0, -36, 0, 1, -50521, 0, 62325, 0, -12810, 0, 1050, 0, -45, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums have e.g.f. exp(x)*sech(x) (signed version of A009006). Inverse of masked Pascal triangle A119467. Transforms the sequence with e.g.f. g(x) to the sequence with e.g.f. g(x)*sech(x).

Coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows). Another version in A153641. - Philippe Deléham, Oct 26 2013

LINKS

Table of n, a(n) for n=0..65.

P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.

Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera

Tom Copeland, Discussion of this sequence

A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414 [Added by Tom Copeland, Aug 31 2015]

P. Luschny, Additive decompositions of classical numbers via the Swiss-Knife polynomils (From Tom Copeland, Oct 20 2015)

Miguel Méndez, Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO}, 2017, Section 4.3, Example 4.

FORMULA

Number triangle whose k-th column has e.g.f. sech(x)*x^k/k!

T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(1/2) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 25 2009

The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n,k)*p{k}(0)*(n mod 2 - 1 + x^(n-k)). - Peter Luschny, Jul 16 2012

E.g.f.: exp(x*z)/cosh(x). - Peter Luschny, Aug 01 2012

Sum_{k=0..n} T(n,k)*x^k = A122045(n), A155585(n), A119880(n), A119881(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 27 2013

EXAMPLE

Triangle begins

1,

0, 1,

-1, 0, 1,

0, -3, 0, 1,

5, 0, -6, 0, 1,

0, 25, 0, -10, 0, 1,

-61, 0, 75, 0, -15, 0, 1,

0, -427, 0, 175, 0, -21, 0, 1,

1385, 0, -1708, 0, 350, 0, -28, 0, 1

MAPLE

T := (n, k) -> binomial(n, k)*2^(n-k)*euler(n-k, 1/2): # Peter Luschny, Jan 25 2009

PROG

(Sage)

@CachedFunction

def A119879_poly(n, x) :

    return 1 if n == 0  else add(A119879_poly(k, 0)*binomial(n, k)*(x^(n-k)-1+n%2) for k in range(n)[::2])

def A119879_row(n) :

    R = PolynomialRing(ZZ, 'x')

    return R(A119879_poly(n, x)).coeffs()  # Peter Luschny, Jul 16 2012

# Alternatively:

(Sage)

# The function riordan_array is defined in A256893.

riordan_array(sech(x), x, 9, exp=true) # Peter Luschny, Apr 19 2015

CROSSREFS

Row sums are A155585. - Johannes W. Meijer, Apr 20 2011

Rows reversed: A081658.

Cf. A109449, A153641, A162660. - Philippe Deléham, Oct 26 2013

Cf. A000182, A046802, A119467, A133314.

Sequence in context: A207543 A191532 A179552 * A115714 A020768 A246182

Adjacent sequences:  A119876 A119877 A119878 * A119880 A119881 A119882

KEYWORD

easy,sign,tabl

AUTHOR

Paul Barry, May 26 2006

STATUS

approved

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Last modified February 21 19:50 EST 2018. Contains 299423 sequences. (Running on oeis4.)