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A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

%I A036829 #38 Jul 26 2025 14:38:19
%S A036829 0,1,7,48,327,2221,15060,102012,690519,4671819,31596447,213633696,
%T A036829 1444131108,9760401756,65957919496,445671648228,3011064814455,
%U A036829 20341769686311,137412453018933,928188965638464,6269358748632207,42343731580741821
%N A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).
%D A036829 M. Petkovsek et al., A=B, Peters, 1996, p. 97.
%H A036829 Reinhard Zumkeller, <a href="/A036829/b036829.txt">Table of n, a(n) for n = 0..1000</a>
%F A036829 G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 09 2011
%F A036829 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 A036829 a(n) ~ 3^(3*n-1)/2^(2*n+1). - _Vaclav Kotesovec_, Dec 29 2012
%F A036829 L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - _Seiichi Manyama_, Jul 19 2025
%F A036829 G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - _Seiichi Manyama_, Jul 26 2025
%t A036829 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 A036829 (Haskell)
%o A036829 a036829 n = sum $ map
%o A036829    (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
%o A036829 -- _Reinhard Zumkeller_, May 24 2012
%Y A036829 Cf. A006256.
%Y A036829 Cf. A001764, A006013, A007318, A036917, A062236.
%Y A036829 Cf. A000302, A308523, A386367, A386368.
%K A036829 nonn
%O A036829 0,3
%A A036829 _N. J. A. Sloane_