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A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).

%I A387008 #19 Aug 27 2025 01:34:41
%S A387008 1,7,46,299,1941,12616,82160,536155,3505699,22964087,150676186,
%T A387008 990134948,6515349244,42925973608,283134975936,1869455684187,
%U A387008 12355133446527,81725384344741,541021064605298,3584203906519219,23761237400402597,157623924396214756,1046244086051121248
%N A387008 a(n) = Sum_{k=0..n} binomial(3*n+3,k).
%H A387008 Vincenzo Librandi, <a href="/A387008/b387008.txt">Table of n, a(n) for n = 0..1000</a>
%F A387008 a(n) = [x^n] (1+x)^(3*n+3)/(1-x).
%F A387008 a(n) = [x^n] 1/((1-x)^(2*n+3) * (1-2*x)).
%F A387008 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+3,k) * binomial(3*n-k+2,n-k).
%F A387008 a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+2,n-k).
%F A387008 G.f.: g^4/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
%F A387008 D-finite with recurrence: 24*(3*n-2)*(3*n-1)*(5*n^2+n-2)*a(n-2) -(295*n^4-156*n^3-339*n^2+12*n+20)*a(n-1) +2*(2*n+1)*(n+1)*(5*n^2-9*n+2)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%F A387008 a(n) ~ 3^(3*n + 7/2) / (sqrt(Pi*n) * 2^(2*n+3)). - _Vaclav Kotesovec_, Aug 20 2025
%t A387008 Table[Sum[Binomial[3*n+3,k], {k,0,n}], {n,0,25}] (* _Vaclav Kotesovec_, Aug 20 2025 *)
%o A387008 (PARI) a(n) = sum(k=0, n, binomial(3*n+3, k));
%o A387008 (Magma) [&+[Binomial(3*n+3,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 27 2025
%Y A387008 Cf. A066380, A160906, A387007.
%Y A387008 Cf. A001764.
%K A387008 nonn,changed
%O A387008 0,2
%A A387008 _Seiichi Manyama_, Aug 12 2025