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 A383254 #33 Aug 19 2025 09:33:45
%S A383254 1,8,51,300,1695,9348,50729,272128,1447155,7643880,40156281,210019428,
%T A383254 1094338401,5684293020,29446107975,152181330480,784880109315,
%U A383254 4040712839880,20768844586025,106595697483700,546389531720445,2797395801163260,14306735857573995
%N A383254 Expansion of 1/sqrt( (1-x) * (1-5*x)^3 ).
%H A383254 Vincenzo Librandi, <a href="/A383254/b383254.txt">Table of n, a(n) for n = 0..600</a>
%F A383254 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+1,k+1).
%F A383254 D-finite with recurrence n*a(n) +2*(-3*n-1)*a(n-1) +5*n*a(n-2)=0. - _R. J. Mathar_, May 05 2025
%F A383254 a(n) ~ 5^(n + 1/2) * sqrt(n/Pi). - _Vaclav Kotesovec_, May 05 2025
%F A383254 From _Seiichi Manyama_, Aug 19 2025: (Start)
%F A383254 a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A383254 a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k) * binomial(n+1,n-k). (End)
%t A383254 Table[Sum[(2*k+1)* Binomial[2*k, k]*Binomial[n+1,k+1],{k,0,n}],{n,0,28}] (* _Vincenzo Librandi_, May 18 2025 *)
%o A383254 (PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+1, k+1));
%o A383254 (Magma) [&+[(2*k+1)*Binomial(2*k, k)*Binomial(n+1, k+1): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, May 18 2025
%Y A383254 Cf. A383499, A383503.
%Y A383254 Cf. A026375.
%K A383254 nonn,easy
%O A383254 0,2
%A A383254 _Seiichi Manyama_, May 05 2025