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A217596 G.f.: x / reversion(x - x^2 - x^3). 1

%I

%S 1,-1,-2,-5,-17,-64,-259,-1098,-4815,-21659,-99385,-463385,-2189070,

%T -10455340,-50402858,-244929608,-1198504743,-5900360016,-29204546125,

%U -145244328630,-725451444795,-3637422742470,-18301949731665,-92380935149100,-467659449093330

%N G.f.: x / reversion(x - x^2 - x^3).

%D H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990.

%H G. C. Greubel, <a href="/A217596/b217596.txt">Table of n, a(n) for n = 0..600</a>

%F a(n) = -Sum_{i=ceiling(n/2)..n} binomial(i,n-i)*binomial(n+i-2,n-2)/(n -1), n>1, a(0)=1, a(1)=-1.

%F a(n) = -Sum_{i=1..n} A001002(n-i)*a(i), a(0)=1.

%F From _Paul D. Hanna_, Mar 19 2013: (Start)

%F G.f. satisfies:

%F (1) A(x) = 1 - x/A(x) - x^2/A(x)^2.

%F (2) A(x - x^2 - x^3) = 1 - x - x^2.

%F (3) [x^n] A(x)^n = -A000204(n), where A000204 is the Lucas numbers.

%F (4) [x^n] A(x)^(n+1) = 0 for n>2. (End)

%F Conjecture: 25*n*(n-1)*a(n) - 5*(n-1)*(25*n-42)*a(n-1) + 3*(-23*n^2 + 59*n + 4)*a(n-2) + 9*(3*n-10)*(3*n-11)*a(n-3)=0. - _R. J. Mathar_, May 23 2014

%F Maple's sumrecursion command gives the second-order recurrence equation: 5*n*(n - 1)*(4*n - 9)*a(n) = 2*(n - 1)*(44*n^2 - 165*n + 150)*a(n-1) + 3*(4*n - 5)(3*n - 7)(3*n - 8)*a(n-2) with initial conditions a(1) = -1 and a(2) = -2. Mathar's conjectured third-order recurrence follows from this. - _Peter Bala_, Feb 15 2015

%e G.f.: A(x) = 1 - x - 2*x^2 - 5*x^3 - 17*x^4 - 64*x^5 - 259*x^6 - 1098*x^7 - ...

%t CoefficientList[x/InverseSeries[Series[x-x^2-x^3, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Feb 15 2015 *)

%o (Maxima)

%o a(n):=if n=0 then 1 else if n=1 then -1 else -sum(binomial(i,n-i)*binomial(n+i-2,n-2),i,ceiling(n/2),n)/(n-1);

%o (PARI) {a(n)=polcoeff(x/serreverse(x-x^2-x^3+x^2*O(x^n)),n)} \\ _Paul D. Hanna_, Mar 19 2013

%o (PARI) /* Using Vladimir Kruchinin's binomial sum: */

%o {a(n)=if(n==0,1,if(n==1,-1,-sum(i=n\2,n,binomial(i,n-i)*binomial(n+i-2,n-2))/(n-1)))} \\ _Paul D. Hanna_, Mar 19 2013

%o (PARI) {a(n)=local(A=1);for(i=1,n,A=1-x/A-x^2/A^2+x*O(x^n));polcoeff(A,n)} \\ _Paul D. Hanna_, Mar 19 2013

%Y Cf. A001002, A007440.

%K sign,easy

%O 0,3

%A _Vladimir Kruchinin_, Mar 19 2013

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Last modified July 17 23:21 EDT 2019. Contains 325109 sequences. (Running on oeis4.)