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A387036 a(n) = Sum_{k=0..n} binomial(4*n-2,k).

%I A387036 #18 Sep 03 2025 01:14:04
%S A387036 1,3,22,176,1471,12616,110056,971712,8656937,77663192,700614760,
%T A387036 6349125440,57754842117,527046644056,4822774262296,44235726874816,
%U A387036 406582639811581,3743845040832376,34529632747211560,318931047174438720,2949641596923575548,27312107861301870368
%N A387036 a(n) = Sum_{k=0..n} binomial(4*n-2,k).
%H A387036 Vincenzo Librandi, <a href="/A387036/b387036.txt">Table of n, a(n) for n = 0..1000</a>
%F A387036 a(n) = [x^n] (1+x)^(4*n-2)/(1-x).
%F A387036 a(n) = [x^n] 1/((1-x)^(3*n-2) * (1-2*x)).
%F A387036 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-2,k) * binomial(4*n-k-3,n-k).
%F A387036 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-3,n-k).
%F A387036 G.f.: 1/(g * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387036 D-finite with recurrence: 128*(4*n-7)*(2*n-3)*(4*n-9)*(22*n^2-17*n-15)*a(n-2) -8*(1892*n^5-11274*n^4+23326*n^3-18132*n^2+1323*n+2835)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^2-61*n+24)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%t A387036 Table[Sum[Binomial[4*n-2,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A387036 (PARI) a(n) = sum(k=0, n, binomial(4*n-2, k));
%o A387036 (Magma) [&+[Binomial(4*n-2, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A387036 Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387037.
%Y A387036 Cf. A002293, A262977.
%K A387036 nonn,changed
%O A387036 0,2
%A A387036 _Seiichi Manyama_, Aug 13 2025