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=10, beta =1 (or alpha=1, beta=10).
In general, for m>=1, C(m; n) ~ m * (4*m)^n / ((2*m - 1)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 10 2019
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..620
FORMULA
G.f.: (1 + 10*x*c(10*x)/9)/(1+x/9) = 1/(1 - x*c(10*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(10^m)/n.
a(n) = (-1/9)^n*(1 - 10*Sum_{k=0..n-1} C(k)*(-90)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*10^(n-k). - Philippe Deléham, Jan 19 2004
a(n) ~ 2^(3*n + 1) * 5^(n+1) / (361*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019
MATHEMATICA
CoefficientList[Series[(19 -Sqrt[1-40*x])/(2*(x+9)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((19 -sqrt(1-40*x))/(2*(x+9))) \\ G. C. Greubel, May 02 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (19 - Sqrt(1-40*x))/(2*(x+9)) )); // G. C. Greubel, May 02 2019
(Sage) ((19 -sqrt(1-40*x))/(2*(x+9))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 13 2001
STATUS
approved