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 A385605 #30 Aug 16 2025 10:02:08
%S A385605 1,7,58,502,4436,39687,358024,3249288,29624796,271080124,2487835678,
%T A385605 22888216006,211010997716,1948830506578,18026768864736,
%U A385605 166976297995452,1548523206590364,14376415735219572,133599985919343400,1242638966005222648,11567295503871866536
%N A385605 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+1,k).
%F A385605 a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n+1)).
%F A385605 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k,k).
%F A385605 a(n) = 3^(4*n+1)*2^(-3*n-1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1/2) - 1). - _Stefano Spezia_, Aug 05 2025
%F A385605 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - _Seiichi Manyama_, Aug 07 2025
%F A385605 G.f.: g^2/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 14 2025
%F A385605 G.f.: B(x)^2/(1 + (B(x)-1)/4), where B(x) is the g.f. of A005810. - _Seiichi Manyama_, Aug 15 2025
%F A385605 G.f.: 1/(1 - x*g^2*(12-5*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 16 2025
%o A385605 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+1, k));
%Y A385605 Cf. A006256, A141223, A385632.
%Y A385605 Cf. A005810, A078995, A147855, A386811.
%Y A385605 Cf. A385498.
%K A385605 nonn
%O A385605 0,2
%A A385605 _Seiichi Manyama_, Aug 03 2025