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A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).

%I A387010 #18 Aug 17 2025 15:48:33
%S A387010 1,8,67,576,5036,44552,397594,3572224,32267668,292750368,2665685155,
%T A387010 24347665728,222972599812,2046626681072,18823260696452,
%U A387010 173427623923712,1600383346290116,14789063407109600,136838247669241276,1267571539176770816,11754134090271100336
%N A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).
%H A387010 Vincenzo Librandi, <a href="/A387010/b387010.txt">Table of n, a(n) for n = 0..1000</a>
%F A387010 a(n) = [x^n] (1+x)^(4*n+3)/(1-x).
%F A387010 a(n) = [x^n] 1/((1-x)^(3*n+3) * (1-2*x)).
%F A387010 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+3,k) * binomial(4*n-k+2,n-k).
%F A387010 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+2,n-k).
%F A387010 G.f.: g^4/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387010 D-finite with recurrence: 128*(4*n-1)*(2*n-1)*(4*n-3)*(11*n^2+8*n+1)*a(n-2) -8*(946*n^5-434*n^4-518*n^3+143*n^2+46*n-3)*a(n-1) +3*n*(3*n+1)*(3*n+2)*(11*n^2-14*n+4)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%t A387010 Table[Sum[Binomial[4*n+3,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 16 2025 *)
%o A387010 (PARI) a(n) = sum(k=0, n, binomial(4*n+3, k));
%o A387010 (Magma) [&+[Binomial(4*n+3, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 16 2025
%Y A387010 Cf. A066381, A386811, A387009, A387011.
%Y A387010 Cf. A002293.
%K A387010 nonn
%O A387010 0,2
%A A387010 _Seiichi Manyama_, Aug 12 2025