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A052129 a(0)=1; thereafter a(n) = n*a(n-1)^2. 17

%I

%S 1,1,2,12,576,1658880,16511297126400,1908360529573854283038720000,

%T 29134719286683212541013468732221146917416153907200000000

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

%C Somos's quadratic recurrence sequence.

%C 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 Unreduced denominators of: f(1) = 1, f(n) = f(n-1) + f(n-1)/(n-1). - _Daniel Suteu_, Jul 29 2016

%D Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159; https://doi.org/10.1016/j.jnt.2016.08.010

%D Chao-Ping Chen, XF Han, On Somos' quadratic recurrence constant, Journal of Number Theory, Volume 166, September 2016, Pages 31-40; http://dx.doi.org/10.1016/j.jnt.2016.02.018

%D S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

%D Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, Vol. 5, No. 1 (April 2011), pp. 60-66; http://www.jstor.org/stable/43666828

%D Xu You, Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, Volume 436, Issue 1, 1 April 2016, Pages 513-520

%H Vincenzo Librandi, <a href="/A052129/b052129.txt">Table of n, a(n) for n = 0..12</a>

%H Sung-Hyuk Cha, <a href="http://csis.pace.edu/~scha/CompComb/CSISTR11-284.pdf">On the k-ary Tree Combinatorics</a>.

%H J. Guillera and J. Sondow, <a href="http://arXiv.org/abs/math.NT/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 J. Guillera and J. Sondow, <a href="http://dx.doi.org/10.1016/10.1007/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 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, Volume 155, October 2015, Pages 36-45.

%H J. Sondow and P. 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 J. Sondow and P. 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>

%F 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 a(n) = n * A030450(n - 1) if n>0. - _Michael Somos_, Oct 22 2006

%F a(n) = (a(n-1) + a(n-2)^2) * (a(n-1) / a(n-2))^2. - _Michael Somos_, Mar 20 2012

%F a(n) = product_{k=1..n} k^(2^(n-k)). - _Jonathan Sondow_, Mar 17 2014

%F A088679(n+1)/a(n) = n+1. -_Daniel Suteu_, Jul 29 2016

%e 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 G.f. = 1 + x + 2*x^2 + 12*x^3 + 576*x^4 + 1658880*x^5 + 16511297126400*x^6 + ...

%t Join[{1},RecurrenceTable[{a[1]==1,a[n]==n a[n-1]^2},a,{n,10}]] (* _Harvey P. Dale_, Apr 26 2011 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], Product[ (n - k)^2^k, {k, 0, n - 1}]]; (* _Michael Somos_, May 24 2013 *)

%t a[n_] := Product[ k^(2^(n - k)), {k,1,n}] (* _Jonathan Sondow_, Mar 17 2014 *)

%t NestList[{#[[1]]+1,#[[1]]*#[[2]]^2}&,{1,1},10][[All,2]] (* _Harvey P. Dale_, Jul 30 2018 *)

%o (PARI) {a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k))}; /* _Michael Somos_, May 24 2013 */

%Y Cf. A000142, A001045, A030450, A112302, A116603, A123851, A123852, A123853, A123854, A238462 (2-adic valuation), A088679.

%K nonn,nice

%O 0,3

%A _Reinhard Zumkeller_, Feb 12 2002

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Last modified December 14 18:21 EST 2018. Contains 318106 sequences. (Running on oeis4.)