cp's OEIS Frontend

A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).

%I A387011 #26 Aug 18 2025 15:42:19
%S A387011 1,9,79,697,6196,55455,499178,4514873,40999516,373585604,3414035527,
%T A387011 31278197839,287191809724,2642070371194,24347999094724,
%U A387011 224723513577529,2076978797223820,19220104372823340,178061257422521452,1651314042800498052,15328459501269535952
%N A387011 a(n) = Sum_{k=0..n} binomial(4*n+4,k).
%H A387011 Vincenzo Librandi, <a href="/A387011/b387011.txt">Table of n, a(n) for n = 0..400</a>
%F A387011 a(n) = [x^n] (1+x)^(4*n+4)/(1-x).
%F A387011 a(n) = [x^n] 1/((1-x)^(3*n+4) * (1-2*x)).
%F A387011 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+4,k) * binomial(4*n-k+3,n-k).
%F A387011 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+3,n-k).
%F A387011 G.f.: g^5/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387011 D-finite with recurrence: 128*(4*n-3)*(2*n+1)*(4*n-1)*(22*n^2+16*n-3)*a(n-2) -8*(1892*n^5+1024*n^4-1982*n^3-1306*n^2-60*n+27)*a(n-1) +3*(n+1)*(3*n+2)*(3*n+1)*(22*n^2-28*n+3)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%F A387011 a(n) ~ 2^(8*n + 15/2) / (sqrt(Pi*n) * 3^(3*n + 7/2)). - _Vaclav Kotesovec_, Aug 18 2025
%t A387011 Table[Sum[Binomial[4*n+4,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 16 2025 *)
%o A387011 (PARI) a(n) = sum(k=0, n, binomial(4*n+4, k));
%o A387011 (Magma) [&+[Binomial(4*n+4, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 16 2025
%Y A387011 Cf. A066381, A386811, A387009, A387010.
%Y A387011 Cf. A002293.
%K A387011 nonn
%O A387011 0,2
%A A387011 _Seiichi Manyama_, Aug 12 2025