A104624 Expansion of exp( arcsinh( -2*x ) ) in powers of x.

1, -2, 2, 0, -2, 0, 4, 0, -10, 0, 28, 0, -84, 0, 264, 0, -858, 0, 2860, 0, -9724, 0, 33592, 0, -117572, 0, 416024, 0, -1485800, 0, 5348880, 0, -19389690, 0, 70715340, 0, -259289580, 0, 955277400, 0, -3534526380, 0, 13128240840, 0, -48932534040, 0, 182965127280, 0, -686119227300, 0

Offset

0,2

Comments

First column in inverse of A054335.

With offset 1 the coefficient sequence of series reversion of A000984 (binomial(2n,n)) also with offset 1. - Michael Somos, Jan 14 2011

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3091.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

G.f.: sqrt( 1 + 4*x^2 ) - 2*x = exp( asinh( -2*x ) ). - Michael Somos, Jan 14 2011

The positive sequence 1,2,2,0,2,... has g.f. 2(1+x)-sqrt(1-4x^2). - Paul Barry, Oct 10 2007

From Vladimir Kruchinin, Jan 16 2011: (Start)

The o.g.f. A(x) satisfies A(x)=x*sqrt(1-4*A(x)),

a(n) = 1/(n*(n+1)) * sum(j=0...n+1, j * 2^(j) * binomial(2*n-j-1,n-1) * binomial(n+1,j) * (-1)^(n-j)). (End)

Conjecture: n*a(n) + (n-1)*a(n-1) + 4*(n-3)*a(n-2) + 4*(n-4)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012

If n is even, a(n) ~ (-1)^(1+n/2) * 2^(n+1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 23 2013

G.f.: 2*S(0) -1-2*x-4*x^2, where S(k) = 2*k+1 + x^2*(2*k+3)/(1 + x^2*(2*k+1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013

a(n) = (-1)^n*2*hypergeom([-n+1, 2-n], [2], -1). - Peter Luschny, Sep 23 2014

Example

G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + 264*x^14 + ...

Maple

s := proc(n) option remember; `if`(n<2, n+1, 4*(n-2)*s(n-2)/(n+1)) end: A104624 := n -> `if`(n<2, (-1)^n*(n+1), (-1)^(n/2-1)*s(n-1)); seq(A104624(n), n=0..47); # Peter Luschny, Sep 23 2014

Mathematica

CoefficientList[ Series[ Exp[ ArcSinh[ -2x]], {x, 0, 49}], x]
Table[(-1)^n 2 HypergeometricPFQ[{-n+1, 2-n}, {2}, -1], {n, 0, 46}] (* Peter Luschny, Sep 23 2014 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) - 2*x, n ) )}; /* Michael Somos, Jan 14 2011 */
(Sage)
def A104624(n):
    if n < 2: return (-1)^n*(n+1)
    if n % 2 == 1: return 0
    return (-1)^(n/2+1)*binomial(n, n/2)/(n-1)
[A104624(n) for n in range(47)] # Peter Luschny, Sep 23 2014
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(-2*x)))); // G. C. Greubel, Aug 12 2018

Crossrefs

Sequence in context: A352564 A353596 A182122 * A371711 A193863 A363566

Adjacent sequences: A104621 A104622 A104623 * A104625 A104626 A104627

Keyword

easy,sign

Author

Paul Barry, Mar 17 2005

Status

approved