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A387034 a(n) = Sum_{k=0..n} binomial(4*n-4,k).

%I A387034 #20 Sep 03 2025 01:13:49
%S A387034 1,1,11,93,794,6885,60460,536155,4791323,43081973,389329652,
%T A387034 3533047572,32174057272,293874981603,2691171713924,24700051833634,
%U A387034 227150464141969,2092620625940629,19308393192688804,178406554524801820,1650535921328322392,15287533448476027572
%N A387034 a(n) = Sum_{k=0..n} binomial(4*n-4,k).
%H A387034 Vincenzo Librandi, <a href="/A387034/b387034.txt">Table of n, a(n) for n = 0..1000</a>
%F A387034 a(n) = [x^n] (1+x)^(4*n-4)/(1-x).
%F A387034 a(n) = [x^n] 1/((1-x)^(3*n-4) * (1-2*x)).
%F A387034 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-4,k) * binomial(4*n-k-5,n-k).
%F A387034 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-5,n-k).
%F A387034 G.f.: 1/(g^3 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387034 D-finite with recurrence: 128*(4*n-11)*(2*n-5)*(4*n-9)*(44*n^3-122*n^2+18*n+105)*a(n-2)-8*(3784*n^6-37684*n^5+141548*n^4-238406*n^3+145758*n^2+37290*n-51975)*a(n-1)+3*n*(3*n-5)*(3*n-7)*(44*n^3-254*n^2+394*n-79)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%F A387034 a(n) ~ 2^(8*n - 17/2) / (sqrt(Pi*n) * 3^(3*n - 9/2)). - _Vaclav Kotesovec_, Aug 20 2025
%t A387034 Table[Sum[Binomial[4*n-4,k], {k,0,n}], {n,0,25}] (* _Vaclav Kotesovec_, Aug 20 2025 *)
%o A387034 (PARI) a(n) = sum(k=0, n, binomial(4*n-4, k));
%o A387034 (Magma) [&+[Binomial(4*n-4, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A387034 Cf. A066381, A386811, A387009, A387010, A387011, A387035, A387036, A387037.
%Y A387034 Cf. A002293, A262977.
%K A387034 nonn,changed
%O A387034 0,3
%A A387034 _Seiichi Manyama_, Aug 13 2025