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A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).

%I A387007 #19 Aug 27 2025 01:35:12
%S A387007 1,6,37,232,1471,9402,60460,390656,2533987,16489546,107594213,
%T A387007 703680424,4611412196,30273024984,199045392232,1310535994368,
%U A387007 8639411571051,57017083602138,376674527189599,2490742704227192,16483857933928471,109175823528400778,723611538997758784
%N A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).
%H A387007 Vincenzo Librandi, <a href="/A387007/b387007.txt">Table of n, a(n) for n = 0..1000</a>
%F A387007 a(n) = [x^n] (1+x)^(3*n+2)/(1-x).
%F A387007 a(n) = [x^n] 1/((1-x)^(2*n+2) * (1-2*x)).
%F A387007 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+2,k) * binomial(3*n-k+1,n-k).
%F A387007 a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+1,n-k).
%F A387007 G.f.: g^3/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
%F A387007 D-finite with recurrence: 24*(5*n+1)*(3*n-1)*(3*n-2)*a(n-2) -(295*n^3-156*n^2-61*n+6)*a(n-1) +2*n*(2*n+1)*(5*n-4)*a(n). - _Georg Fischer_, Aug 17 2025
%t A387007 Table[Sum[Binomial[3*n+2,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 27 2025 *)
%o A387007 (PARI) a(n) = sum(k=0, n, binomial(3*n+2, k));
%o A387007 (Magma) [&+[Binomial(3*n+2,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 27 2025
%Y A387007 Cf. A066380, A160906, A387008.
%Y A387007 Cf. A001764.
%K A387007 nonn,changed
%O A387007 0,2
%A A387007 _Seiichi Manyama_, Aug 12 2025