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a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2*k) * binomial(2*k,k).
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%I #13 Sep 20 2024 15:35:54

%S 1,0,-2,-2,4,10,-4,-38,-22,114,188,-234,-914,-18,3376,3338,-9416,

%T -21718,14416,96338,39274,-328558,-471344,795398,2586064,-517690,

%U -10453424,-8272658,32186818,63596494,-61876584,-307070174,-62655330,1129250706,1356328788

%N a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-1-k,n-2*k) * binomial(2*k,k).

%H Harvey P. Dale, <a href="/A360313/b360313.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 / sqrt(1+4*x^2/(1-x)).

%F n*a(n) = 2*(n-1)*a(n-1) - (5*n-6)*a(n-2) + 2*(2*n-5)*a(n-3).

%t RecurrenceTable[{a[0]==1,a[1]==0,a[2]==-2,a[n]==1/n (2(n-1)a[n-1]-(5n-6)a[n-2]+2(2n-5)a[n-3])},a,{n,40}] (* _Harvey P. Dale_, Sep 20 2024 *)

%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(n-1-k, n-2*k)*binomial(2*k, k));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt(1+4*x^2/(1-x)))

%Y Cf. A360314, A360315.

%Y Cf. A026585, A360293.

%K sign

%O 0,3

%A _Seiichi Manyama_, Feb 03 2023