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 A386812 #40 Aug 21 2025 07:20:11
%S A386812 1,7,67,697,7547,83682,942649,10739176,123388763,1427090845,
%T A386812 16593192942,193774331494,2271115189673,26700463884244,
%U A386812 314735943548632,3718522618187472,44021808206431579,522080025971331983,6201449551502245321,73767447652621434695,878599223738760686422
%N A386812 a(n) = Sum_{k=0..n} binomial(5*n+1,k).
%H A386812 Vincenzo Librandi, <a href="/A386812/b386812.txt">Table of n, a(n) for n = 0..500</a>
%F A386812 a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n+1)).
%F A386812 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k,k).
%F A386812 D-finite with recurrence 8*n*(2754528070303487*n -4672004545621835)*(4*n-3)*(2*n-1) *(4*n-1)*a(n) +(-5828620079131711179*n^5 -135826272187971586019*n^4 +779361612339655552281*n^3 -1570139520911413863589*n^2 +1419656431480813021170*n -487668485184225269400)*a(n-1) +40*(-21123668262204329085*n^5 +243394620512022153401*n^4 -982249084763267479011*n^3 +1849334401749026834935*n^2 -1662134287466221884960*n +573649997457991096080)*a(n-2) +6400*(5*n-13)*(5*n-11)*(2475036532470005*n-2376524337096748)*(5*n-9)*(5*n-12)*a(n-3)=0. - _R. J. Mathar_, Aug 03 2025
%F A386812 a(n) = 2^(5*n+1) - binomial(5*n+1, n)*(hypergeom([1, -1-4*n], [1+n], -1) - 1). - _Stefano Spezia_, Aug 05 2025
%F A386812 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k). - _Seiichi Manyama_, Aug 07 2025
%F A386812 G.f.: g^2/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 12 2025
%F A386812 From _Seiichi Manyama_, Aug 16 2025: (Start)
%F A386812 G.f.: 1/(1 - x*g^3*(10-3*g)) where g = 1+x*g^5 is the g.f. of A002294.
%F A386812 G.f.: B(x)^2/(1 + 3*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)
%F A386812 a(n) ~ 5^(5*n + 3/2) / (3*sqrt(Pi*n) * 2^(8*n + 3/2)). - _Vaclav Kotesovec_, Aug 21 2025
%t A386812 Table[Sum[Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 21 2025 *)
%o A386812 (PARI) a(n) = sum(k=0, n, binomial(5*n+1, k));
%o A386812 (Magma) [&+[Binomial(5*n+1, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 21 2025
%Y A386812 Cf. A000302, A160906, A386811.
%Y A386812 Cf. A002294, A371739.
%Y A386812 Cf. A001449, A079589, A079678, A371753, A385632.
%K A386812 nonn
%O A386812 0,2
%A A386812 _Seiichi Manyama_, Aug 03 2025