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%I #14 Mar 04 2014 13:45:31
%S 1,1,14,203,3038,46746,736764,11853051,194053622,3224557406,
%T 54265836548,923218762270,15854602773100,274500192707860,
%U 4786546243533432,83989334625037947,1481965556616225702
%N Generalized Catalan numbers.
%C a(n) = K(7,7; n)/7 with K(a,b; n) defined in a comment to A068763.
%H Fung Lam, <a href="/A068769/b068769.txt">Table of n, a(n) for n = 0..780</a>
%F a(n) = (7^n) * p(n, -6/7) with the row polynomials p(n, x) defined from array A068763.
%F a(n+1) = 7*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
%F G.f.: (1-sqrt(1-28*x*(1-6*x)))/(14*x).
%F Recurrence: (n+1)*a(n) = 168*(2-n)*a(n-2) + 14*(2*n-1)*a(n-1). - _Fung Lam_, Mar 04 2014
%F a(n) ~ sqrt(14+14*sqrt(7)) * (14+2*sqrt(7))^n / (14*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 04 2014
%t a[0] = 1; a[1] = 1; a[2] = 14; a[n_] := (168 (2 - n) a[n - 2] + 14 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 04 2014 *)
%t CoefficientList[Series[(1-Sqrt[1-28*x*(1-6*x)])/(14*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%Y Cf. A000108, A068764-A068768, A068770-A068772, A025227-A025230.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 04 2002
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