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a(n) = Sum_{k=0..n} C(n,3k+2)^2.
4

%I #21 Sep 08 2022 08:45:33

%S 0,0,1,9,36,101,261,882,3921,17253,67554,243695,876789,3324906,

%T 13166791,52301709,203824548,782913717,3010327497,11695756698,

%U 45823049817,179787741723,703527078258,2747647985241,10739885115573,42082084255050,165225573240651

%N a(n) = Sum_{k=0..n} C(n,3k+2)^2.

%C The recurrence is same as for A119363. - _Vaclav Kotesovec_, Mar 12 2019

%H Seiichi Manyama, <a href="/A139469/b139469.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 4^n / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 12 2019

%t Table[Sum[Binomial[n, 3*k + 2]^2, {k, 0, n}], {n, 0, 40}] (* _Vaclav Kotesovec_, Mar 12 2019 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n, 3*k+2)^2); \\ _Michel Marcus_, Mar 12 2019

%o (Magma) [&+[Binomial(n, 3*k+2)^2: k in [0..n]]: n in [0..30]]; // _Vincenzo Librandi_, Mar 14 2019

%Y Cf. A024495, A119363, A139468.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Jun 12 2008