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a(1)=0, a(2)=1; for n>2, a(n) = C(n,2)*(1+a(n-2)).
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%I #30 Sep 08 2022 08:45:29

%S 0,1,3,12,40,195,861,5488,31032,247005,1706815,16302396,133131648,

%T 1483518127,13978823145,178022175360,1901119947856,27237392830233,

%U 325091511083547,5175104637744460,68269217327545080,1195449171318970491

%N a(1)=0, a(2)=1; for n>2, a(n) = C(n,2)*(1+a(n-2)).

%C a(n) is also the maximum number of ways to place node pairs in an area formed by n 1 X 1 squares. - _Theodore M. Mishura_, Mar 20 2015

%F a(n) = A087214(n) - 1. - _N. J. A. Sloane_, Feb 15 2007

%F a(n) = Sum_{k=1..floor(n/2)} 2^k*Pochhammer(-n/2,k)*Pochhammer(1/2-n/2,k). - _Theodore M. Mishura_, Mar 16 2015

%F a(n) ~ n! * (exp(sqrt(2)) + (-1)^n * exp(-sqrt(2))) / 2^(n/2+1). - _Vaclav Kotesovec_, Mar 20 2015

%p seq(simplify(hypergeom([1, 1-n/2, 3/2-n/2], [], 2))*(n-1)*n/2,n=1..22); # _Mark van Hoeij_, May 12 2013

%t nxt[{n_,a_,b_}]:={n+1,b,Binomial[n+1,2](a+1)}; Transpose[NestList[nxt,{2,0,1},30]][[2]] (* _Harvey P. Dale_, Oct 12 2014 *)

%o (Magma) I:=[0,1]; [n le 2 select I[n] else Binomial(n,2)*(1+Self(n-2)): n in [1..35]]; // _Vincenzo Librandi_, Mar 17 2015

%Y Cf. A087214.

%K nonn

%O 1,3

%A Allan L. Edmonds (edmonds(AT)indiana.edu), Feb 13 2007

%E Edited by _Vladeta Jovovic_, Feb 20 2009