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A000651
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Running time of Takeuchi function.
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2
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0, 1, 4, 14, 53, 223, 1034, 5221, 28437, 165859, 1029803, 6772850, 46983238, 342509396, 2615606677, 20865444825, 173446634597, 1499111445237, 13445550920288, 124919896067530, 1200320663197275, 11910845573790488
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OFFSET
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0,3
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REFERENCES
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D. E. Knuth, personal communication.
V. Lifschitz, editor, Artificial intelligence and mathematical theory of computation. Papers in honor of John McCarthy. Academic Press, Inc., Boston, MA, 1991. See p. 215.
T. Prellberg, On the asymptotics of Takeuchi numbers, Symbolic computation, number theory, special functions, physics and combinatorics, Kluwer Acad. Publ., Dordrecht, 2001, pp. 231-242. MR 2002m:11016.
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LINKS
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FORMULA
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G.f. A(z) satisfies A(z-z^2)/z - A(z) = 1/(1-z) + z/(1-z+z^2). (Prellberg).
Asymptotic growth: a(n) ~ C_T*B(n)*exp(1/2*W(n)^2), where B(n) are the Bell numbers, W(n) the Lambert W function and C_T = 2.2394331040...(Prellberg).
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MATHEMATICA
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a[n_] := a[n] = If[n < 1, 0, Sum[ (2*k)!/k!/(k+1)!, {k, 1, n}] + Sum[ (2*Binomial[n+k-1, k] - Binomial[n+k, k])*a[n-1-k], {k, 0, n-2}]]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2013, after Pari *)
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PROG
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(PARI) a(n)=if(n<1, 0, sum(k=1, n, (2*k)!/k!/(k+1)!)+sum(k=0, n-2, (2*binomial(n+k-1, k)-binomial(n+k, k))*a(n-1-k)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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