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A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

%I A377011 #73 Sep 03 2025 04:23:52
%S A377011 1,6,34,188,1026,5556,29940,160824,862018,4613636,24667644,131795912,
%T A377011 703812916,3757135752,20051429544,106992663408,570827898306,
%U A377011 3045193326372,16244056119084,86646747723048,462161936699196,2465043081687192,13147597801986264,70123266087502608
%N A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).
%H A377011 Vincenzo Librandi, <a href="/A377011/b377011.txt">Table of n, a(n) for n = 0..1000</a>
%F A377011 a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
%F A377011 a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
%F A377011 a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
%F A377011 G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
%F A377011 a(n) ~ 2^(4*n+2) / 3^(n+1). - _Vaclav Kotesovec_, Aug 20 2025
%F A377011 D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - _R. J. Mathar_, Aug 21 2025
%t A377011 Table[Sum[3^k * Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A377011 (PARI) a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));
%o A377011 (Magma) [&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A377011 Cf. A293490, A383832, A386942.
%Y A377011 Cf. A000302, A000984, A026641, A141223, A386957.
%K A377011 nonn,changed
%O A377011 0,2
%A A377011 _Seiichi Manyama_, Aug 11 2025