OFFSET
0,3
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.8, p. 321.
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
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
T. Prellberg, On the asymptotic analysis of a class of linear recurrences (slides).
T. Prellberg, On the Asymptotics of Takeuchi Numbers, arXiv:math/0005008 [math.CO], 2000.
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
KEYWORD
nonn,changed
AUTHOR
EXTENSIONS
Typo in formula corrected by Vaclav Kotesovec, Sep 16 2013
STATUS
approved