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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 4.
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%I #10 Jan 25 2015 16:41:23

%S 1,-1,1,3,4,3,4,19,60,124,214,455,1266,3404,7974,17699,42050,107956,

%T 276534,680140,1647872,4082562,10366604,26363583,66318590,166265630,

%U 420151570,1070885100,2734066540,6964626146,17739885228,45334834067,116258144838

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 4.

%D A. Cayley, On the Porism of the In-and-circumscribed Polygon, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 4, 292-308 (see p. 300).

%F G.f.: (1-sqrt(1-4*x+8*x^2-12*x^3))/2.

%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 8*(n-3)*a(n-2) + 6*(2*n-9)*a(n-3). - _Vaclav Kotesovec_, Jan 25 2015

%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = -1; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n-k]],{k,1,n-1}],{n,4,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-4*x+8*x^2-12*x^3+x*O(x^n)))/2,n)

%K sign

%O 1,4

%A _Clark Kimberling_

%E Additional comments from _Michael Somos_, Apr 19, 2000