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A387035 a(n) = Sum_{k=0..n} binomial(4*n-3,k).

%I A387035 #18 Sep 03 2025 01:14:42
%S A387035 1,2,16,130,1093,9402,82160,726206,6474541,58115146,524472448,
%T A387035 4754293704,43257431931,394821713910,3613377083248,33146854168628,
%U A387035 304692552429413,2805871076597738,25880523571338272,239058748663208600,2211058130414688244,20474163633488699944
%N A387035 a(n) = Sum_{k=0..n} binomial(4*n-3,k).
%H A387035 Vincenzo Librandi, <a href="/A387035/b387035.txt">Table of n, a(n) for n = 0..1000</a>
%F A387035 a(n) = [x^n] (1+x)^(4*n-3)/(1-x).
%F A387035 a(n) = [x^n] 1/((1-x)^(3*n-3) * (1-2*x)).
%F A387035 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-3,k) * binomial(4*n-k-4,n-k).
%F A387035 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-4,n-k).
%F A387035 G.f.: 1/(g^2 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387035 D-finite with recurrence: 128*(4*n-7)*(2*n-5)*(4*n-9)*(22*n^3-50*n^2+5*n+30)*a(n-2) -8*(1892*n^6-16004*n^5+51038*n^4-73470*n^3+39874*n^2+6165*n-9450)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^3-116*n^2+171*n-47)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%t A387035 Table[Sum[Binomial[4*n-3,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A387035 (PARI) a(n) = sum(k=0, n, binomial(4*n-3, k));
%o A387035 (Magma) [&+[Binomial(4*n-3, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A387035 Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387036, A387037.
%Y A387035 Cf. A002293, A262977.
%K A387035 nonn,changed
%O A387035 0,2
%A A387035 _Seiichi Manyama_, Aug 13 2025