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A025246
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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-3)*a(3) for n >= 4.
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7
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1, 0, 1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 212, 438, 910, 1903, 4009, 8494, 18080, 38656, 82988, 178802, 386490, 837928, 1821664, 3970282, 8673258, 18987930, 41652382, 91539466, 201525238, 444379907, 981384125, 2170416738, 4806513660, 10657780276
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OFFSET
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1,6
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1+x-sqrt(1-2*x+x^2-4*x^3))/2. - Michael Somos, Jun 08 2000
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
a(n) = hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, 3 - n], 27) for n >= 3. - Peter Luschny, Jun 15 2022
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MAPLE
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a := n -> ifelse(n < 3, 0^(n - 1),
hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, -n + 3], 27)):
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MATHEMATICA
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a[n_]:= a[n]= If[n<4, 1-Boole[n==2], Sum[a[j]*a[n-j], {j, n-3}]];
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PROG
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(PARI) a(n)=polcoeff((1+x-sqrt(1-2*x+x^2-4*x^3+x*O(x^n)))/2, n)
(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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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