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A064310
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Generalized Catalan numbers C(-1; n).
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10
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1, 1, 0, 1, -2, 6, -18, 57, -186, 622, -2120, 7338, -25724, 91144, -325878, 1174281, -4260282, 15548694, -57048048, 210295326, -778483932, 2892818244, -10786724388, 40347919626, -151355847012, 569274150156
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OFFSET
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0,5
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COMMENTS
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See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
Unsigned sequence with a(0) := 0 is A000957 (Fine).
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REFERENCES
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Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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a(n)= sum((n-m)*binomial(n-1+m, m)*((-1)^m)/n, m=0..n-1) = ((1/2)^n)*(1+sum(C(k)*(-2)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+x*c(-x)/2)/(1-x/2) = 1/(1-x*c(-x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{k, 0<=k<=n} (-1)^(n-k)*A106566(n, k) . - Philippe DELEHAM, Sep 18 2005
(-1)^n*a(n)=Sum_{k, 0<=k<=n}A039599(n,k)*(-2)^k . - Philippe DELEHAM, Mar 13 2007
Conjecture: 2*n*a(n) +(7*n-12)*a(n-1) +2*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Dec 02 2012
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CROSSREFS
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Sequence in context: A125306 A209797 * A126983 A104629 A000957 A125305
Adjacent sequences: A064307 A064308 A064309 * A064311 A064312 A064313
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KEYWORD
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sign,easy
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AUTHOR
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Wolfdieter Lang, Sep 21 2001
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STATUS
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approved
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