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a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).
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%I #26 May 10 2020 03:00:51

%S 0,1,7,48,327,2221,15060,102012,690519,4671819,31596447,213633696,

%T 1444131108,9760401756,65957919496,445671648228,3011064814455,

%U 20341769686311,137412453018933,928188965638464,6269358748632207,42343731580741821

%N a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

%D M. Petkovsek et al., A=B, Peters, 1996, p. 97.

%H Reinhard Zumkeller, <a href="/A036829/b036829.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 09 2011

%F Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - _Vaclav Kotesovec_, Nov 19 2012

%F a(n) ~ 3^(3*n-1)/2^(2*n+1). - _Vaclav Kotesovec_, Dec 29 2012

%t Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Jan 10 2012 *)

%o (Haskell)

%o a036829 n = sum $ map

%o (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]

%o -- _Reinhard Zumkeller_, May 24 2012

%Y Cf. A006256.

%Y Cf. A007318, A036917.

%K nonn

%O 0,3

%A _N. J. A. Sloane_