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A052129 a(0) = 1; thereafter a(n) = n*a(n-1)^2.

%I A052129 #121 Feb 16 2025 08:32:41
%S A052129 1,1,2,12,576,1658880,16511297126400,1908360529573854283038720000,
%T A052129 29134719286683212541013468732221146917416153907200000000
%N A052129 a(0) = 1; thereafter a(n) = n*a(n-1)^2.
%C A052129 Somos's quadratic recurrence sequence.
%C A052129 Iff n is prime (n>2), the n-adic valuation of a(2n) is 3*A001045(n) (three times the values at the prime indices of Jacobsthal numbers), which is 2^n+1. For example: the 11-adic valuation at a(22) = 2049 = 3*A001045(11)= 683.  3*683 = 2^11+1 = 2049.  True because: When n is prime, n-adic valuation is 1 at A052129(n), then doubles as n-increases to 2n, at which point 1 is added; thus A052129(2n) = 2^n+1. Since 3*A001045(n) = 2^n+1, n-adic valuation of A052129(2n) = 3*A001045(n) when n is prime. - _Bob Selcoe_, Mar 06 2014
%C A052129 Unreduced denominators of: f(1) = 1, f(n) = f(n-1) + f(n-1)/(n-1). - _Daniel Suteu_, Jul 29 2016
%D A052129 S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
%H A052129 Vincenzo Librandi, <a href="/A052129/b052129.txt">Table of n, a(n) for n = 0..12</a>
%H A052129 Sung-Hyuk Cha, <a href="http://csis.pace.edu/~scha/CompComb/CSISTR11-284.pdf">On the k-ary Tree Combinatorics</a>.
%H A052129 Chao-Ping Chen, <a href="https://doi.org/10.1016/j.jnt.2016.08.010">Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant</a>, Journal of Number Theory, 172 (2017), 145-159.
%H A052129 Chao-Ping Chen and X. F. Han, <a href="http://dx.doi.org/10.1016/j.jnt.2016.02.018">On Somos' quadratic recurrence constant</a>, Journal of Number Theory, Volume 166, September 2016, Pages 31-40.
%H A052129 Olivier Golinelli, <a href="https://arxiv.org/abs/2405.16968">Remote control system of a binary tree of switches - II. balancing for a perfect binary tree</a>, arXiv:2405.16968 [cs.DM], 2024. See p. 16.
%H A052129 Jesús Guillera and Jonathan Sondow, <a href="https://arxiv.org/abs/math/0506319">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, arXiv:math/0506319 [math.NT], 2005-2006.
%H A052129 Jesús Guillera and Jonathan Sondow, <a href="https://doi.org/10.1007/s11139-007-9102-0">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, Ramanujan J. 16 (2008), 247-270.
%H A052129 Dawei Lu and Zexi Song, <a href="http://dx.doi.org/10.1016/j.jnt.2015.03.013">Some new continued fraction estimates of the Somos' quadratic recurrence constant</a>, Journal of Number Theory, 155 (2015), 36-45.
%H A052129 Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, <a href="https://doi.org/10.1007/s00025-018-0928-0">Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant</a>, Results in Mathematics 74(1) (2019), Article 6.
%H A052129 Gergo Nemes, <a href="http://www.jstor.org/stable/43666828">On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant</a>, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
%H A052129 Jörg Neunhäuserer, <a href="https://arxiv.org/abs/2006.02882">On the universality of Somos' constant</a>, arXiv:2006.02882 [math.DS], 2020.
%H A052129 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:math/0610499 [math.CA], 2006.
%H A052129 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 A052129 Xu You and Di-Rong Chen, <a href="https://doi.org/10.1016/j.jmaa.2015.12.013">Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant</a>, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
%H A052129 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>.
%F A052129 a(n) ~ s^(2^n) / (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where s = 1.661687949633... (see A112302) and A116603. - _Michael Somos_, Apr 02 2006
%F A052129 a(n) = n * A030450(n - 1) if n>0. - _Michael Somos_, Oct 22 2006
%F A052129 a(n) = (a(n-1) + a(n-2)^2) * (a(n-1) / a(n-2))^2. - _Michael Somos_, Mar 20 2012
%F A052129 a(n) = product_{k=1..n} k^(2^(n-k)). - _Jonathan Sondow_, Mar 17 2014
%F A052129 A088679(n+1)/a(n) = n+1. -_Daniel Suteu_, Jul 29 2016
%e A052129 a(3) = 3*a(2)^2 = 3*(2*a(1)^2)^2 = 3*(2*(1*a(0)^2)^2)^2 = 3*(2*(1*1^2)^2)^2 = 3*(2*1)^2 = 3*4 = 12.
%e A052129 G.f. = 1 + x + 2*x^2 + 12*x^3 + 576*x^4 + 1658880*x^5 + 16511297126400*x^6 + ...
%t A052129 Join[{1},RecurrenceTable[{a[1]==1,a[n]==n a[n-1]^2},a,{n,10}]]  (* _Harvey P. Dale_, Apr 26 2011 *)
%t A052129 a[ n_] := If[ n < 1, Boole[n == 0], Product[ (n - k)^2^k, {k, 0, n - 1}]]; (* _Michael Somos_, May 24 2013 *)
%t A052129 a[n_] := Product[ k^(2^(n - k)), {k,1,n}] (* _Jonathan Sondow_, Mar 17 2014 *)
%t A052129 NestList[{#[[1]]+1,#[[1]]*#[[2]]^2}&,{1,1},10][[All,2]] (* _Harvey P. Dale_, Jul 30 2018 *)
%o A052129 (PARI) {a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k))}; /* _Michael Somos_, May 24 2013 */
%Y A052129 Cf. A000142, A001045, A030450, A088679, A112302, A116603, A123851, A123852, A123853, A123854, A238462 (2-adic valuation).
%K A052129 nonn,nice
%O A052129 0,3
%A A052129 _Reinhard Zumkeller_, Feb 12 2002