This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227035 #23 Oct 06 2021 12:57:25
%S A227035 1,1,1,1,2,6,16,36,76,172,436,1156,3006,7606,19202,49466,130156,
%T A227035 345356,915196,2421532,6427001,17163581,46087911,124133531,334850208,
%U A227035 904691576,2449891276,6651540676,18100561856,49344295152,134719523056,368350942416,1008680051756
%N A227035 a(n) = Sum_{k=0..floor(n/4)} binomial(n,4*k)*binomial(5*k,k)/(4*k+1).
%C A227035 Generally, Sum(binomial(n,p*k)*binomial((p+1)*k,k)/(p*k+1), k=0..floor(n/p)) is asymptotic to (p+(p+1)^(1+1/p))^(n+3/2)/(p^(n+1)*(p+1)^(1+3/(2*p))*n^(3/2)*sqrt(2*Pi)).
%H A227035 Vincenzo Librandi, <a href="/A227035/b227035.txt">Table of n, a(n) for n = 0..1000</a>
%F A227035 Recurrence: -2869*(n-7)*(n-6)*(n-5)*(n-4)*a(n-8) + 2*(n-6)*(n-5)*(n-4)*(5226*n-17267)*a(n-7) - (n-5)*(n-4)*(11582*n^2-55156*n+50139)*a(n-6) - 3*(n-4)*(612*n^3 - 18926*n^2 + 102684*n - 155665)*a(n-5) + 5*(n-4)*(2959*n^3 - 26172*n^2 + 77408*n - 76800)*a(n-4) - 1024*(n-2)*(2*n-5)*(7*n^2-35*n+48)*a(n-3) + 1024*(n-2)*(n-1)*(7*n^2-28*n+30)*a(n-2) - 1024*(n-2)*(n-1)*n*(2*n-3)*a(n-1) + 256*(n-2)*(n-1)*n*(n+1)*a(n) = 0.
%F A227035 a(n) ~ (4+5^(1+1/4))^(n+3/2)/(4^(n+1)*5^(1+3/8)*n^(3/2)*sqrt(2*Pi)).
%F A227035 G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^5. - _Ilya Gutkovskiy_, Jul 25 2021
%F A227035 From _Peter Bala_, Sep 15 2021: (Start)
%F A227035 O.g.f.: A(x) = (1/x)*series reversion ( x*(1 - x^4)/(1 + x*(1 - x^4) )).
%F A227035 The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion ( x*(1 - x^4)/(1 + (m + 1)*x*(1 - x^4)) ). The case m = -1 gives the sequence [1,0,0,0,1,0,0,0,5,0,0,0,35,0,0,0,285,...] - an aerated version of A002294. (End)
%t A227035 Table[Sum[Binomial[n,4*k]*Binomial[5*k,k]/(4*k+1),{k,0,Floor[n/4]}],{n,0,20}]
%o A227035 (PARI) a(n)=sum(k=0,n\4,binomial(n,4*k)*binomial(5*k,k)/(4*k+1)) \\ _Charles R Greathouse IV_, Jun 28 2013
%Y A227035 Cf. A002294, A007317 (p=1), A049130 (p=2), A226974 (p=3), A226910 (p=5).
%K A227035 nonn,easy
%O A227035 0,5
%A A227035 _Vaclav Kotesovec_, Jun 28 2013