A168076 Expansion of 1 - 3*(1-x-sqrt(1-2*x-3*x^2))/2.

1, 0, -3, -3, -6, -12, -27, -63, -153, -381, -969, -2505, -6564, -17394, -46533, -125505, -340902, -931716, -2560401, -7070337, -19609146, -54597852, -152556057, -427642677, -1202289669, -3389281245, -9578183391, -27130207503, -77009455428, -219023318406

Offset: 0

Author

Paul Barry, Nov 18 2009

Keywords

• easy
• sign

Comments

For n>0, a(n) = -3*A168049(n). Hankel transform is A168075. Another variant is A168073.

Links

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

Crossrefs

Cf. A168049, A168073, A168075.

Programs

Maple

f:= gfun:-rectoproc({(3*n-3)*a(n)+(1+2*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 0, a(2) = -3},a(n),remember):
map(f, [$0..60]); # Robert Israel, May 13 2018

Mathematica

CoefficientList[Series[1 - 3*(1 - x - Sqrt[1 - 2*x - 3*x^2])/2, {x,0,50}] , x] (* G. C. Greubel, Jul 09 2016 *)

PARI

x='x+O('x^99); Vec(1-3*(1-x-(1-2*x-3*x^2)^(1/2))/2) \\ Altug Alkan, May 13 2018

PARI

A168076(n)=!n-3*sum(k=0,n\2-1, binomial(n-2,2*k)*binomial(2*k,k)/(k+1)) \\ M. F. Hasler, May 13 2018

Formula

a(n) = 0^n - 3*Sum_{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.

D-finite with recurrence: n*a(n) + (-2*n+3)*a(n-1) + 3*(-n+3)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014

Recurrence (for n >= 3) follows from the differential equation (3*x^2+2*x-1)*y' - (3*x+1)*y = 3*x-1 satisfied by the g.f. - Robert Israel, May 13 2018

a(n) ~ -3^(n+1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 03 2014

a(n) = -A168073(n) <= 0 for n >= 1. - M. F. Hasler, May 13 2018

Extensions

Comment corrected by Vaclav Kotesovec, Dec 03 2014