Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Sep 08 2022 08:45:04
%S 1,1,-6,85,-1490,29226,-614004,13511709,-307448490,7174776190,
%T -170777485556,4130050311234,-101192982385844,2506610481299380,
%U -62668163792277840,1579300030107459885,-40076101342241993370
%N Generalized Catalan numbers C(-7; n).
%C See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%H G. C. Greubel, <a href="/A064329/b064329.txt">Table of n, a(n) for n = 0..690</a>
%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-7)^m/n.
%F a(n) = (1/8)^n*(1 + 7*Sum_{k=0..n-1} C(k)*(-7*8)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F G.f.: (1+7*x*c(-7*x)/8)/(1-x/8) = 1/(1-x*c(-7*x)) with c(x) g.f. of Catalan numbers A000108.
%t CoefficientList[Series[(15 +Sqrt[1+28*x])/(2*(8-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((15 +sqrt(1+28*x))/(2*(8-x))) \\ _G. C. Greubel_, May 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (15 +Sqrt(1+28*x))/(2*(8-x)) )); // _G. C. Greubel_, May 03 2019
%o (Sage) ((15 +sqrt(1+28*x))/(2*(8-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 03 2019
%K sign,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 21 2001