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A387009 a(n) = Sum_{k=0..n} binomial(4*n+2,k).

%I A387009 #19 Aug 17 2025 15:46:15
%S A387009 1,7,56,470,4048,35443,313912,2804012,25211936,227881004,2068564064,
%T A387009 18844224462,172186125456,1577401391626,14483100716176,
%U A387009 133240186921816,1227901991526976,11333497984085620,104752914242685856,969417048912326008,8981452266787224128
%N A387009 a(n) = Sum_{k=0..n} binomial(4*n+2,k).
%H A387009 Vincenzo Librandi, <a href="/A387009/b387009.txt">Table of n, a(n) for n = 0..1000</a>
%F A387009 a(n) = [x^n] (1+x)^(4*n+2)/(1-x).
%F A387009 a(n) = [x^n] 1/((1-x)^(3*n+2) * (1-2*x)).
%F A387009 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+2,k) * binomial(4*n-k+1,n-k).
%F A387009 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+1,n-k).
%F A387009 G.f.: g^3/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387009 D-finite with recurrence: 128*(4*n-3)*(2*n-1)*(4*n-5)*(22*n+5)*a(n-2) -8*(1892*n^4-3706*n^3+1750*n^2+214*n-177)*a(n-1) +3*(22*n-17)*(n-1)*(3*n-1)*(3*n+1)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%t A387009 Table[Sum[Binomial[4*n+2,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 16 2025 *)
%o A387009 (PARI) a(n) = sum(k=0, n, binomial(4*n+2, k));
%o A387009 (Magma) [&+[Binomial(4*n+2, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 16 2025
%Y A387009 Cf. A066381, A386811, A387010, A387011.
%Y A387009 Cf. A002293.
%K A387009 nonn
%O A387009 0,2
%A A387009 _Seiichi Manyama_, Aug 12 2025