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 A386834 #16 Aug 19 2025 03:37:28
%S A386834 1,8,111,1738,28701,488412,8473387,148994510,2645999673,47349481408,
%T A386834 852429930567,15421507805106,280126256513109,5105764838932388,
%U A386834 93331970924544099,1710369544783134614,31412304686874624113,578023658034894471048,10654486069487503147135
%N A386834 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k-1,n-k).
%F A386834 a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n).
%F A386834 a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(3*n)).
%F A386834 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(4*n+1,k).
%F A386834 a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(3*n+k-1,k).
%F A386834 Conjecture D-finite with recurrence +3*n*(16543753*n -26995933)*(3*n-1)*(3*n-2)*a(n) +(-1669899251*n^4 -26931977989*n^3 +131963667975*n^2 -188283072995*n +85757456660)*a(n-1) +2*(-61301926003*n^4 +515926265010*n^3 -1655392333929*n^2 +2311146075302*n -1165379619540)*a(n-2) -96*(39221117*n -50949760)*(4*n-9)*(2*n-5)*(4*n-7)*a(n-3)=0. - _R. J. Mathar_, Aug 19 2025
%o A386834 (PARI) a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(4*n-k-1, n-k));
%Y A386834 Cf. A116881, A386833.
%Y A386834 Cf. A370101, A386837.
%Y A386834 Cf. A383716.
%K A386834 nonn
%O A386834 0,2
%A A386834 _Seiichi Manyama_, Aug 05 2025