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Search: id:A064087
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| A064087 |
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Generalized Catalan numbers C(4; n). |
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+0
5
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| 1, 1, 5, 41, 413, 4641, 55797, 702297, 9137549, 121909457, 1658755685, 22929591433, 321111942781, 4546112358529, 64958195967957, 935566629270201, 13567825195172973, 197957440018622769, 2903721563443327557
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n+1)= Y_{n}(n+1)= Z_{n} in the Derrida et al. 1992 reference (see A064094) for alpha=4, beta =1 (or alpha=1, beta=4).
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FORMULA
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G.f.: (1+4*x*c(4*x)/3)/(1+x/3) = 1/(1-x*c(4*x)) with c(x) g.f. of Catalan numbers A000108.
a(n)= sum((n-m)*binomial(n-1+m, m)*(4^m)/n, m=0..n-1) = ((-1/3)^n)*(1-4*sum(C(k)*(-12)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum{ k= 0...n, A059365(n, k)*4^(n-k) } . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-3*x^2)/(1+x)^2+O(x^(n+1))), n)) (from R. Stephan)
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CROSSREFS
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A064063 (C(3, n)).
Sequence in context: A083884 A026000 A058475 this_sequence A081215 A140095 A083073
Adjacent sequences: A064084 A064085 A064086 this_sequence A064088 A064089 A064090
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001
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