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A025246 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.

%I A025246 #28 Jan 16 2025 11:15:40
%S A025246 1,0,1,1,1,2,4,7,13,26,52,104,212,438,910,1903,4009,8494,18080,38656,
%T A025246 82988,178802,386490,837928,1821664,3970282,8673258,18987930,41652382,
%U A025246 91539466,201525238,444379907,981384125,2170416738,4806513660,10657780276
%N A025246 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.
%C A025246 Essentially the same as A023431.
%H A025246 G. C. Greubel, <a href="/A025246/b025246.txt">Table of n, a(n) for n = 1..1000</a>
%H A025246 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=666">Encyclopedia of Combinatorial Structures 666</a>
%F A025246 a(n) = A023431(n-3).
%F A025246 G.f.: (1+x-sqrt(1-2*x+x^2-4*x^3))/2. - _Michael Somos_, Jun 08 2000
%F A025246 n*a(n) = (2*n-3)*a(n-1) -(n-3)*a(n-2) +2*(2*n-9)*a(n-3). - _R. J. Mathar_, Feb 25 2015
%F A025246 a(n) = hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, 3 - n], 27) for n >= 3. - _Peter Luschny_, Jun 15 2022
%p A025246 a := n -> ifelse(n < 3, 0^(n - 1),
%p A025246    hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, -n + 3], 27)):
%p A025246 seq(simplify(a(n)), n = 1..32); # _Peter Luschny_, Jun 15 2022
%t A025246 a[n_]:= a[n]= If[n<4, 1-Boole[n==2], Sum[a[j]*a[n-j], {j,n-3}]];
%t A025246 Table[a[n], {n, 45}] (* _G. C. Greubel_, Jun 15 2022 *)
%o A025246 (PARI) a(n)=polcoeff((1+x-sqrt(1-2*x+x^2-4*x^3+x*O(x^n)))/2,n)
%o A025246 (Magma) [n le 2 select 2-n else (&+[Binomial(n-k-3, 2*k)*Catalan(k): k in [0..Floor((n-3)/3)]]): n in [1..45]]; // _G. C. Greubel_, Jun 15 2022
%o A025246 (SageMath) [bool(n==1) + sum(binomial(n-k-3,2*k)*catalan_number(k) for k in (0..((n-3)//3))) for n in (1..45)] # _G. C. Greubel_, Jun 15 2022
%Y A025246 Cf. A000108, A023431.
%K A025246 nonn
%O A025246 1,6
%A A025246 _Clark Kimberling_