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A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

%I A386811 #44 Aug 21 2025 09:57:58
%S A386811 1,6,46,378,3214,27896,245506,2182396,19548046,176142312,1594831736,
%T A386811 14497410186,132224930146,1209397179048,11088872706188,
%U A386811 101890087382168,937973964234638,8649109175873288,79872298511230120,738583466508887304,6837944227813170424
%N A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).
%H A386811 Vincenzo Librandi, <a href="/A386811/b386811.txt">Table of n, a(n) for n = 0..500</a>
%F A386811 a(n) = [x^n] 1/((1-2*x) * (1-x)^(3*n+1)).
%F A386811 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k,k).
%F A386811 D-finite with recurrence +645*n*(3*n-1)*(3*n-2)*a(n) +8*(-56722*n^3+213090*n^2-305978*n+150255)*a(n-1) +128*(62908*n^3-282348*n^2+385070*n-126735)*a(n-2) +12288*(-2486*n^3+8918*n^2+758*n-18935)*a(n-3) -2949120*(2*n-7)*(4*n-13)*(4*n-11)*a(n-4)=0. - _R. J. Mathar_, Aug 03 2025
%F A386811 a(n) = 2^(4*n+1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1) - 1). - _Stefano Spezia_, Aug 05 2025
%F A386811 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - _Seiichi Manyama_, Aug 07 2025
%F A386811 a(n) ~ 2^(8*n + 3/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). - _Vaclav Kotesovec_, Aug 07 2025
%F A386811 G.f.: g^2/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 12 2025
%F A386811 G.f.: B(x)^2/(1 + (B(x)-1)/2), where B(x) is the g.f. of A005810. - _Seiichi Manyama_, Aug 15 2025
%F A386811 G.f.: 1/(1 - x*g^2*(8-2*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 16 2025
%t A386811 Table[Sum[Binomial[4*n+1,k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 07 2025 *)
%o A386811 (PARI) a(n) = sum(k=0, n, binomial(4*n+1, k));
%o A386811 (Magma) [&+[Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 21 2025
%Y A386811 Cf. A000302, A160906, A386812.
%Y A386811 Cf. A005810, A078995, A147855, A226733, A226761, A385605.
%Y A386811 Cf. A098430, A383716.
%Y A386811 Cf. A066381.
%K A386811 nonn
%O A386811 0,2
%A A386811 _Seiichi Manyama_, Aug 03 2025