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A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

%I A387085 #30 Aug 31 2025 12:47:22
%S A387085 1,0,4,8,36,120,456,1680,6340,23960,91224,348656,1337896,5149872,
%T A387085 19877904,76907808,298176516,1158168792,4505865144,17555689008,
%U A387085 68490100536,267518448912,1046041377264,4094231982048,16039426479336,62887835652720,246761907761776,968943740083040
%N A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).
%H A387085 Vincenzo Librandi, <a href="/A387085/b387085.txt">Table of n, a(n) for n = 0..1000</a>
%F A387085 a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
%F A387085 a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
%F A387085 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
%F A387085 a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
%F A387085 G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
%F A387085 G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
%F A387085 G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
%F A387085 G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
%F A387085 D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - _R. J. Mathar_, Aug 19 2025
%t A387085 Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 31 2025 *)
%o A387085 (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));
%o A387085 (Magma) [&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 31 2025
%Y A387085 Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
%Y A387085 Cf. A226733, A226705, A226751.
%Y A387085 Cf. A005810, A079589, A183160.
%K A387085 nonn,changed
%O A387085 0,3
%A A387085 _Seiichi Manyama_, Aug 16 2025