A099252 Bisection of A005043.

0, 1, 6, 36, 232, 1585, 11298, 83097, 625992, 4805595, 37458330, 295673994, 2358641376, 18985057351, 154000562758, 1257643249140, 10331450919456, 85317692667643, 707854577312178, 5897493615536452, 49320944483427000, 413887836110423787, 3484084625456932134, 29412628894558563849

Offset: 0

Author

N. J. A. Sloane, Nov 16 2004

Keywords

• nonn
• easy

References

• G. F. Smith, On isotropic tensors and rotation tensors of dimension m and order n, Tensor (N.S.), Vol. 19 (1968), 79-88 (MR0224008).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.

Crossrefs

Cf. A005043, A099251.

Programs

Maple

G := (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)): Gser := series(G,x=0,60):
seq(coeff(Gser, x^(2*n-1)), n=1..25); # Emeric Deutsch
a := n -> -hypergeom([-2*n-1, 1/2], [2], 4):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Jul 26 2020

Mathematica

Take[CoefficientList[Series[(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x)), {x, 0, 60}], x], {2, -1, 2}] (* Vaclav Kotesovec, Oct 17 2012 *)

PARI

x='x+O('x^66); v=Vec((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x))); vector(#v\2,n,v[2*n]) \\ Joerg Arndt, May 12 2013

Sage

def A():
    a, b, c, d, n = 0, 1, 1, -1, 1
    yield 0
    while True:
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
        c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
        if n%2: yield -(d + b)*(1-(-1)^n)//2
A099252  = A()
print([next(A099252) for  in range(24)]) # _Peter Luschny, May 16 2016

Formula

Recurrence: (n+1)*(2*n+1)*a(n) = n*(26*n-7)*a(n-1) - 3*(26*n^2 - 61*n + 39)*a(n-2) + 27*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 3^(2*n+5/2)/(16*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

a(n) = -hypergeom([-2*n - 1, 1/2], [2], 4). - Peter Luschny, Jul 26 2020

Extensions

More terms from Emeric Deutsch, Nov 18 2004