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 A248586 #28 May 25 2025 11:04:51
%S A248586 1,5,45,521,6733,92385,1316865,19274925,287694285,4359037985,
%T A248586 66837293545,1034774126325,16149186405025,253737607849445,
%U A248586 4009771017244485,63681603585696321,1015763347140335565,16264070907887454465
%N A248586 a(n) = Sum_{i=0..n} C(n,i)*C(2i,i)^2.
%H A248586 Seiichi Manyama, <a href="/A248586/b248586.txt">Table of n, a(n) for n = 0..815</a>
%F A248586 a(n) = Sum_{i=0..n} A007318(n,i)*A002894(i).
%F A248586 Conjecture: n^2*a(n) +(-19*n^2+19*n-5)*a(n-1) +35*(n-1)^2*a(n-2) -17*(n-1)*(n-2)*a(n-3)=0.
%F A248586 G.f.: LegendreP(-1/2, (1+15x)/(1-17x)) /[sqrt(1-17x)*sqrt(1-x)]. [Corrected by _Robert Israel_, Oct 28 2016]
%F A248586 From _Emanuele Munarini_, Oct 28 2016: (Start)
%F A248586 a(n) = hypergeometric(1/2,1/2,-n;1,1;-16).
%F A248586 G.f.: A(t) = (2/Pi)*(ellipticK(16*t/(1-t))/(1-t)).
%F A248586 Diff. eq. satisfied by the g.f.: t*(1-t)*(1-18*t+17*t^2)*A''(t)+(1-t)*(1-37*t+68*t^2)*A'(t)-(34*t^2-35*t+5)*A(t)=0.
%F A248586 Remark: the conjectured recurrence for the coefficients a(n) comes from this diff. eq. for A(t).
%F A248586 (End)
%F A248586 a(n) ~ 17^(n+1)/(16*Pi*n). - _Vaclav Kotesovec_, Oct 30 2016
%t A248586 Table[Sum[Binomial[n, k] Binomial[2k, k]^2, {k, 0, n}],{n,0,100}] (* _Emanuele Munarini_, Oct 28 2016 *)
%o A248586 (PARI) a(n) = sum(i=0, n, binomial(n,i)*binomial(2*i,i)^2); \\ _Michel Marcus_, Oct 09 2014
%o A248586 (Maxima) makelist(sum(binomial(n,k)*binomial(2*k,k)^2,k,0,n),n,0,12); /* _Emanuele Munarini_, Oct 28 2016 */
%Y A248586 Cf. A002894 (inverse binomial transform), A002893.
%K A248586 nonn,easy
%O A248586 0,2
%A A248586 _R. J. Mathar_, Oct 09 2014