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%I #47 Sep 08 2022 08:45:28
%S 1,1,2,24,55296,845378412871680,
%T 3624972460853492659595005581182702601633792000
%N A cubic recurrence: a(0) = 1, a(n) = n*a(n-1)^3 for n >= 1.
%C A cubic analog of Somos's quadratic recurrence sequence A052129.
%C Terms a(7) onward are too big to include in data section. - _G. C. Greubel_, Aug 10 2019
%D S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
%H G. C. Greubel, <a href="/A123851/b123851.txt">Table of n, a(n) for n = 0..8</a>
%H Sung-Hyuk Cha, <a href="http://csis.pace.edu/~scha/CompComb/CSISTR11-284.pdf">On the k-ary Tree Combinatorics</a>.
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://arxiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:0610499 [math.CA], 2006.
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl. 332 (2007), 292-314.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>.
%H Aimin Xu, <a href="https://doi.org/10.1142/S1793042119501112">Asymptotic expansion related to the Generalized Somos Recurrence constant</a>, International Journal of Number Theory 15(10) (2019), 2043-2055.
%F a(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.
%e a(3) = 3*a(2)^3 = 3*(2*a(1)^3)^3 = 3*(2*(1*a(0)^3)^3)^3 = 3*(2*(1*1^3)^3)^3 = 3*(2*1)^3 = 3*8 = 24.
%e G.f. = 1 + x + 2*x^2 + 24*x^3 + 55296*x^4 + 845378412871680*x^5 + ...
%t a[n_]:= If[n==0, 1, n*a[n-1]^3]; Table[a[n], {n,0,7}]
%t nxt[{n_,a_}]:={n+1, (n+1)a^3}; NestList[nxt,{0,1},7][[All,2]] (* _Harvey P. Dale_, May 25 2019 *)
%o (PARI) {a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^3^k))}; /* _Michael Somos_, Aug 07 2016 */
%o (Magma) [n eq 0 select 1 else (&*[(n-k)^(3^k): k in [0..n-1]]):n in [0..8]]; // _G. C. Greubel_, Aug 10 2019
%o (Sage) [1]+[prod((n-k)^(3^k) for k in (0..n-1)) for n in (1..8)] # _G. C. Greubel_, Aug 10 2019
%o (GAP) List([0..8], n-> Product([0..n-1], k-> (n-k)^(3^k)) ); # _G. C. Greubel_, Aug 10 2019
%Y Cf. A052129, A112302, A116603, A123852, A123853, A123854.
%K easy,nonn
%O 0,3
%A _Petros Hadjicostas_ and _Jonathan Sondow_, Oct 15 2006
%E Corrected by _Harvey P. Dale_, May 25 2019
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