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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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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FORMULA
| 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 (kolotoko(AT)wanadoo.fr), Sep 18 2005
(-1)^n*a(n)=Sum_{k, 0<=k<=n}A039599(n,k)*(-2)^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 13 2007
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CROSSREFS
| Sequence in context: A071721 A125306 * A126983 A104629 A000957 A125305
Adjacent sequences: A064307 A064308 A064309 * A064311 A064312 A064313
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KEYWORD
| sign,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 21 2001
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