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A064088 Generalized Catalan numbers C(5; n). 9
1, 1, 6, 61, 766, 10746, 161376, 2537781, 41260086, 687927166, 11698135396, 202104763026, 3537486504556, 62595852983236, 1117926476207316, 20124876291104421, 364797768048805926, 6652740911381353206 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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).
LINKS
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_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(5^m)/n.
a(n) = (-1/4)^n*(1 - 5*Sum_{k=0..n-1} C(k)*(-20)^k, n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*5^(n-k). - Philippe Deléham, Jan 19 2004
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)
Conjecture: 4*n*a(n) +(-79*n+120)*a(n-1) +10*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) ~ 4^n * 5^(n+1) / (81*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019
MATHEMATICA
a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*5^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(9-Sqrt[1-20*x])/(2*(x+4)), {x, 0, 30}], x] (* G. C. Greubel, May 02 2019 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-4*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9-Sqrt(1-20*x))/(2*(x+4)) )); // G. C. Greubel, May 02 2019
(Sage) ( (9-sqrt(1-20*x))/(2*(x+4)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019
CROSSREFS
Cf. A064087 (C(4, n)).
Sequence in context: A142970 A034659 A369509 * A342110 A191803 A259271
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 13 2001
STATUS
approved

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Last modified June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)