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

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006256.

Cf. A001764, A006013, A007318, A036917, A062236.

Cf. A000302, A308523, A386367, A386368.

Programs

Haskell

a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012

Mathematica

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 *)

Formula

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011

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

a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012

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

G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025