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A240688 Expansion of -(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1). 1
1, 1, 5, 19, 81, 351, 1553, 6959, 31489, 143551, 658305, 3033471, 14034177, 65147135, 303285505, 1415422719, 6620053505, 31021657087, 145613977601, 684537354239, 3222408929281, 15187861143551, 71663163121665 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

FORMULA

a(n) = Sum_{k=0..n} Sum_{i=0..(n-k)} binomial(k,n-k-i)*binomial(k+i-1,i))*binomial(n,k)).

A(x) = x*D'(x)/D(x) where D(x)=(1-sqrt(1-4*x-4*x^2))/(2*(1+x)) is g.f. of A052709.

a(n) ~ 2^(n-1/4) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 12 2014

a(n) = sum(i=0..n/2, binomial(n,i)*binomial(2*n-2*i-1,n-2*i)). - Vladimir Kruchinin, Mar 10 2015

Conjecture: n*(n-1)*a(n) -(3*n-2)*(n-1)*a(n-1) +2*(-4*n^2+7*n-1)*a(n-2) -4*n*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016

MATHEMATICA

CoefficientList[Series[-(x Sqrt[-4 x^2 - 4 x + 1] - 2 x^2 - 3 x) / ((x + 1) Sqrt[-4 x^2 - 4 x + 1] + 4 x^3 + 8 x^2 + 3 x - 1), {x, 0, 25}], x] (* Vaclav Kotesovec, Apr 12 2014 *)

PROG

(Maxima)

a(n):=sum((sum(binomial(k, n-k-i)*binomial(k+i-1, i), i, 0, n-k))*binomial(n, k), k, 0, n);

(PARI) x='x+O('x^50); Vec(-(x*sqrt(-4*x^2-4*x+1)-2*x^2-3*x) / ((x+1)*sqrt(-4*x^2-4*x+1)+ 4*x^3+8*x^2+3*x-1)) \\ G. C. Greubel, Apr 05 2017

CROSSREFS

Cf. A052709.

Sequence in context: A149782 A149783 A149784 * A149785 A149786 A149787

Adjacent sequences:  A240685 A240686 A240687 * A240689 A240690 A240691

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, Apr 10 2014

STATUS

approved

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Last modified November 14 15:08 EST 2019. Contains 329126 sequences. (Running on oeis4.)