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A064088
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Generalized Catalan numbers C(5; n).
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4
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1, 1, 6, 61, 766, 10746, 161376, 2537781, 41260086, 687927166, 11698135396, 202104763026, 3537486504556, 62595852983236, 1117926476207316, 20124876291104421, 364797768048805926, 6652740911381353206
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=5, beta =1 (or alpha=1, beta=5).
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FORMULA
| G.f.: (1+5*x*c(5*x)/4)/(1+x/4) = 1/(1-x*c(5*x)) with c(x) g.f. of Catalan numbers A000108.
a(n)= sum((n-m)*binomial(n-1+m, m)*(5^m)/n, m=0..n-1) = ((-1/4)^n)*(1-5*sum(C(k)*(-20)^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)*5^(n-k) } . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 19 2004
Contribution from Gary W. Adamson, Jul 18 2011: (Start)
a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0,...
5, 5, 5, 0, 0, 0,...
5, 5, 5, 5, 0, 0,...
5, 5, 5, 5, 5, 0,...
5, 5, 5, 5, 5, 5,...
... (end)
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-4*x^2)/(1+x)^2+O(x^(n+1))), n)) (from R. Stephan)
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CROSSREFS
| A064087 (C(4, n))
Sequence in context: A160751 A142970 A034659 * A191803 A047737 A086403
Adjacent sequences: A064085 A064086 A064087 * A064089 A064090 A064091
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 13 2001
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