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 A225612 #30 Apr 02 2017 04:06:37
%S A225612 1,5,33,253,2073,17577,152173,1336213,11854513,105997793,953658321,
%T A225612 8622997453,78291531921,713305091521,6518037055321,59712126248041,
%U A225612 548239063327621,5043390644753269,46475480410336709,428936432074181109,3964252574286355429
%N A225612 Partial sums of the binomial coefficients C(4*n,n).
%C A225612 Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.
%H A225612 G. C. Greubel, <a href="/A225612/b225612.txt">Table of n, a(n) for n = 0..1000</a>
%F A225612 Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).
%F A225612 a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).
%p A225612 A225612:=n->add(binomial(4*k,k), k=0..n): seq(A225612(n), n=0..30); # _Wesley Ivan Hurt_, Apr 01 2017
%t A225612 Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
%t A225612 Accumulate[Table[Binomial[4n,n],{n,0,20}]] (* _Harvey P. Dale_, Feb 01 2015 *)
%o A225612 (PARI) for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ _G. C. Greubel_, Apr 01 2017
%Y A225612 Cf. A006134 (p=2), A188675 (p=3), A225615 (p=5).
%K A225612 nonn,easy
%O A225612 0,2
%A A225612 _Vaclav Kotesovec_, Aug 06 2013