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A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).

%I A385497 #51 Aug 26 2025 07:12:49
%S A385497 1,8,92,1160,15276,206368,2835200,39419864,553000876,7811733392,
%T A385497 110962066532,1583318009160,22677731944032,325849065291056,
%U A385497 4694837606889424,67803714186207280,981265566082447276,14227018304102548368,206608052310739404392,3004777578508008253808
%N A385497 a(n) = Sum_{k=0..n} binomial(6*n+1,k).
%H A385497 Vincenzo Librandi, <a href="/A385497/b385497.txt">Table of n, a(n) for n = 0..500</a>
%F A385497 a(n) = [x^n] (1+x)^(6*n+1)/(1-x).
%F A385497 a(n) = [x^n] 1/((1-x)^(5*n+1) * (1-2*x)).
%F A385497 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
%F A385497 a(n) = Sum_{k=0..n} 2^k * binomial(6*n-k,n-k).
%F A385497 G.f.: 1/(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
%F A385497 G.f.: g^2/((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
%F A385497 G.f.: B(x)^2/(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.
%F A385497 a(n) ~ 2^(6*n-1) * 3^(6*n + 3/2) / (sqrt(Pi*n) * 5^(5*n + 1/2)). - _Vaclav Kotesovec_, Aug 19 2025
%F A385497 D-finite with recurrence +5*n*(5*n-3) *(25275337086729240289198339046875*n +471647298106881091699147254457046) *(5*n-1)*(5*n-4)*(5*n-2)*a(n) +(78985428396028875903744809521484375*n^6 -559942234844855804767211877804090453801*n^5 +3587636672285250929619857349305543417315*n^4 -10153151347942687598200945831585305558855*n^3 +14794114656715293872778407292185015920550*n^2 -10846691360081598422810600143797325763664*n +3179147242764665659301361496311050364480)*a(n-1) +40*(916451705547792050816664342989042382392*n^6 -15754440652132350078674083937326518806004*n^5 +117614110896134855700514819789186651267682*n^4 -471111363407608954402735569277858473721059*n^3 +1053743992048348087929158710510276422876431*n^2 -1242809524683997363700671579060256757555078*n +603414490131980309336751304501155726403152) *a(n-2) +3072*(-950768355029313182341332806167821761828*n^6 +17097100921628721474237101055297828968024*n^5 -128090998271831890487248970509140383514230*n^4 +509544263618626898681417576914870842148685*n^3 -1132270964907780344616429736070172799129247*n^2 +1330655887974191637410201798934319046990726*n -645481184978535641217111809931780144149880) *a(n-3) +884736*(3*n-11) *(6*n-17) *(61801507754400081418308631750717123*n -123657551673181017806623428016627104) *(6*n-19)*(3*n-10)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Aug 26 2025
%t A385497 Table[Sum[Binomial[6*n+1,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 18 2025 *)
%o A385497 (PARI) a(n) = sum(k=0, n, binomial(6*n+1, k));
%o A385497 (Magma) [&+[Binomial(6*n+1, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 18 2025
%Y A385497 Cf. A000302, A160906, A386811, A386812.
%Y A385497 Cf. A004355, A079590, A079679, A226705.
%Y A385497 Cf. A002295, A385719.
%K A385497 nonn,changed
%O A385497 0,2
%A A385497 _Seiichi Manyama_, Aug 17 2025