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 A385319 #26 Aug 01 2025 07:49:39
%S A385319 1,5,43,422,4387,47090,515854,5731052,64330531,727812026,8285505178,
%T A385319 94798502804,1089146648206,12556967516852,145201851788092,
%U A385319 1683334752235352,19558532125813027,227694254392461962,2655343386035416162,31014205667706302852,362746369474101224602
%N A385319 a(n) = Sum_{k=0..n} 2^k * binomial(2*n,k) * binomial(2*n-k-1,n-k).
%F A385319 a(n) = [x^n] ( (1+2*x)^2/(1-x) )^n.
%F A385319 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^2 ). See A371391.
%F A385319 a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(2*n,k).
%F A385319 a(n) ~ 2^(2*n-2) * 3^(n+1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Jul 31 2025
%F A385319 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+k-1,k). - _Seiichi Manyama_, Aug 01 2025
%t A385319 Table[Sum[3^k*(-1)^(n-k)*Binomial[2*n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 31 2025 *)
%o A385319 (PARI) a(n) = sum(k=0, n, 2^k*binomial(2*n,k)*binomial(2*n-k-1, n-k));
%Y A385319 Cf. A385320, A386719.
%Y A385319 Cf. A371391.
%K A385319 nonn
%O A385319 0,2
%A A385319 _Seiichi Manyama_, Jul 31 2025