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A217596
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G.f.: x / reversion(x - x^2 - x^3).
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1
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1, -1, -2, -5, -17, -64, -259, -1098, -4815, -21659, -99385, -463385, -2189070, -10455340, -50402858, -244929608, -1198504743, -5900360016, -29204546125, -145244328630, -725451444795, -3637422742470, -18301949731665, -92380935149100, -467659449093330
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OFFSET
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0,3
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REFERENCES
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H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990.
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LINKS
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FORMULA
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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.
a(n) = -Sum_{i=1..n} A001002(n-i)*a(i), a(0)=1.
G.f. satisfies:
(1) A(x) = 1 - x/A(x) - x^2/A(x)^2.
(2) A(x - x^2 - x^3) = 1 - x - x^2.
(4) [x^n] A(x)^(n+1) = 0 for n>2. (End)
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
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
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EXAMPLE
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G.f.: A(x) = 1 - x - 2*x^2 - 5*x^3 - 17*x^4 - 64*x^5 - 259*x^6 - 1098*x^7 - ...
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MATHEMATICA
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CoefficientList[x/InverseSeries[Series[x-x^2-x^3, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Feb 15 2015 *)
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PROG
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(Maxima)
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);
(PARI) {a(n)=polcoeff(x/serreverse(x-x^2-x^3+x^2*O(x^n)), n)} \\ Paul D. Hanna, Mar 19 2013
(PARI) /* Using Vladimir Kruchinin's binomial sum: */
{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
(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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