A271318 Expansion of 1/(-x*sqrt(4*x^2+1)-x^2+1).

1, 1, 2, 5, 9, 16, 33, 67, 126, 239, 477, 946, 1809, 3471, 6858, 13515, 26001, 50100, 98577, 193705, 373734, 721691, 1416933, 2779780, 5372001, 10386841, 20366802, 39915887, 77216409, 149422384, 292749633, 573363693, 1109898126

Offset

0,3

Comments

a(365) is negative. - Vaclav Kotesovec, Apr 04 2016

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} Sum_{j=floor(n/2)..(n+k)/2} 4^(j-k)*binomial(k,2*j-n)*binomial((2*j-n)/2,j-k).

D-finite with recurrence: (n-1)*a(n) = -(n-13)*a(n-2) + 3*(5*n-17)*a(n-4) + 12*(n-4)*a(n-6). - Vaclav Kotesovec, Apr 04 2016

a(n) ~ (1 + 1/sqrt(21))/2 * ((3 + sqrt(21))/2)^(n/2) if n is even and a(n) ~ (-1)^((n+1)/2) * 2^(n+7/2) / (25*sqrt(Pi)*n^(3/2)) if n is odd. - Vaclav Kotesovec, Apr 04 2016

Mathematica

CoefficientList[Series[1/(-x*Sqrt[4*x^2+1]-x^2+1), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 04 2016 *)

Prog

(Maxima) a(n):=sum(sum(4^(j-k)*binomial(k, 2*j-n)*binomial((2*j-n)/2, j-k), j, floor(n/2), (n+k)/2), k, 0, n);
(PARI) x='x+O('x^99); Vec(1/(-x*sqrt(4*x^2+1)-x^2+1)) \\ Altug Alkan, Apr 04 2016

Crossrefs

Sequence in context: A130752 A309331 A059529 * A119676 A348836 A036711

Adjacent sequences: A271315 A271316 A271317 * A271319 A271320 A271321

Keyword

sign,easy

Author

Vladimir Kruchinin, Apr 04 2016

Status

approved