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 A000651 Running time of Takeuchi function. 2
 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, 165859, 1029803, 6772850, 46983238, 342509396, 2615606677, 20865444825, 173446634597, 1499111445237, 13445550920288, 124919896067530, 1200320663197275, 11910845573790488 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES 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. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..570 P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490, 2011 T. Prellberg, On the asymptotic analysis of a class of linear recurrences (slides). T. Prellberg, On the Asymptotics of Takeuchi Numbers Eric Weisstein's World of Mathematics, Takeuchi Number FORMULA 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). MATHEMATICA 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 *) PROG (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))) CROSSREFS Cf. A143307. Sequence in context: A112872 A162482 A308555 * A192247 A118896 A145211 Adjacent sequences:  A000648 A000649 A000650 * A000652 A000653 A000654 KEYWORD nonn AUTHOR EXTENSIONS Typo in formula corrected by Vaclav Kotesovec, Sep 16 2013 STATUS approved

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Last modified February 29 05:01 EST 2020. Contains 332353 sequences. (Running on oeis4.)