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A052129 a(n) = if n>0 then n*a(n-1)^2 else 1. 17
1, 1, 2, 12, 576, 1658880, 16511297126400, 1908360529573854283038720000, 29134719286683212541013468732221146917416153907200000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Somos's quadratic recurrence sequence.

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 A520129(n), then doubles as n-increases to 2n, at which point 1 is added; thus A520129(2n) = 2^n+1. Since 3*A001045(n) = 2^n+1, n-adic valuation of A520129(2n) = 3*A001045(n) when n is prime. - Bob Selcoe, Mar 06 2014

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..12

Sung-Hyuk Cha, On the k-ary Tree Combinatorics.

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007) 292-314.

Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant

FORMULA

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). - Michael Somos, Apr 02 2006

a(n) = n * A030450(n - 1) if n>0. - Michael Somos, Oct 22 2006

a(n) = (a(n-1) + a(n-2)^2) * (a(n-1) / a(n-2))^2. - Michael Somos, Mar 20 2012

a(n) = product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 17 2014

EXAMPLE

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.

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

MATHEMATICA

Join[{1}, RecurrenceTable[{a[1]==1, a[n]==n a[n-1]^2}, a, {n, 10}]]  (* Harvey P. Dale, Apr 26 2011 *)

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

LinearRecurrence[{#1*#2}, {1}, 9] (* Robert G. Wilson v, Jun 15 2013 *)

a[n_] := Product[ k^(2^(n - k)), {k, 1, n}] (* Jonathan Sondow, Mar 17 2014 *)

PROG

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

CROSSREFS

Cf. A000142, A001045, A030450, A112302, A116603, A123851, A123852, A123853, A123854, A238462 (2-valuations).

Sequence in context: A145513 A002860 A108078 * A216335 A173104 A141770

Adjacent sequences:  A052126 A052127 A052128 * A052130 A052131 A052132

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Feb 12 2002

STATUS

approved

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Last modified April 24 10:55 EDT 2014. Contains 240983 sequences.