A123851 -id:A123851 - cp's OEIS frontend

A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984, 288230376151711744, 1152921504606846976

Offset: 0

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Numerators are A123853.
Equals 2^A004134(n); also the denominators in expansion of (1-x)^(-1/4). - Alexander Adamchuk, Oct 27 2006
All terms are powers of 2 and log_2 a(n) = A004134(n) = 3*n - A000120(n). - Alexander Adamchuk, Oct 27 2006 [Edited by Petros Hadjicostas, May 14 2020]
Is this the same sequence as A088802? - N. J. A. Sloane, Mar 21 2007
Almost certainly this is the same as A088802. - Michael Somos, Aug 23 2007
Denominators of Gegenbauer_C(2n,1/4,2). The denominators of Gegenbauer_C(n,1/4,2) give the doubled sequence. - Paul Barry, Apr 21 2009
If the Greubel formula in A088802 and the Luschny formula here are correct (they are the same), the sequence is a duplicate of A088802. - R. J. Mathar, Aug 02 2023

			A123851(n) ~ c^(3^n)*n^(- 1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros 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.
Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]

Cf. A052129, A112302, A116603, A123851, A123852, A123853 (numerators).

Cf. A004134, A004130, A000120.

f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do denom(coeff(series(f(3,x),x=0,30),x,j)); od;
# Alternatively:
A123854 := n -> denom(binomial(1/4,n)):
seq(A123854(n), n=0..25); # Peter Luschny, Apr 07 2016

Denominator[CoefficientList[Series[ 1/Sqrt[Sqrt[1-x]], {x, 0, 25}], x]] (* Robert G. Wilson v, Mar 23 2014 *)

vector(25, n, n--; denominator(binomial(1/4,n)) ) \\ G. C. Greubel, Aug 08 2019

# uses[A000120]
def A123854(n): return 1 << (3*n-A000120(n))
[A123854(n) for n in (0..25)]  # Peter Luschny, Dec 02 2012

From Alexander Adamchuk, Oct 27 2006: (Start)
a(n) = 2^A004134(n).
a(n) = 2^(3n - A000120(n)). (End)
a(n) = denominator(binomial(1/4,n)). - Peter Luschny, Apr 07 2016

A123853 Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.

Original entry on oeis.org

1, 3, -15, 113, -5397, 84813, -3267755, 74391561, -15633072909, 465681118929, -31041303829713, 1145088996404679, -185348722911971841, 8165727090278785521, -778296382754673737187, 39898888480559205453945, -35033447016186321707305533

Offset: 0

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Denominators are A123854.

			A123851(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

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

T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros 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.
Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]

Cf. A052129, A112302, A116603, A123851, A123852, A123854 (denominators).

f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do numer(coeff(series(f(3,x),x=0,30),x,j)); od;

{a(n) = local(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x * O(x^k); A += x^k * polcoeff( 3/4 * (subst(1/A, x, x^2/(1-x^2))^2/(1-x^2) - 1/subst(A, x, x^2)^(2/3)), 2*k ) ); numerator( polcoeff( A, n ) ) ) } /* Michael Somos, Aug 23 2007 */

A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

Original entry on oeis.org

1, 6, 6, 1, 6, 8, 7, 9, 4, 9, 6, 3, 3, 5, 9, 4, 1, 2, 1, 2, 9, 5, 8, 1, 8, 9, 2, 2, 7, 4, 9, 9, 5, 0, 7, 4, 9, 9, 6, 4, 4, 1, 8, 6, 3, 5, 0, 2, 5, 0, 6, 8, 2, 0, 8, 1, 8, 9, 7, 1, 1, 1, 6, 8, 0, 2, 5, 6, 0, 9, 0, 2, 9, 8, 2, 6, 3, 8, 3, 7, 2, 7, 9, 0, 8, 3, 6, 9, 1, 7, 6, 4, 1, 1, 4, 6, 1, 1, 6, 7, 1, 5, 5, 2, 8

Offset: 1

Michael Somos, Sep 02 2005

From Johannes W. Meijer, Jun 27 2016: (Start)
With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.
For similar formulas, see A090998 and A135002. For background information, see A274181.
The structure of the n! * LP(n) formulas leads to the multinomial coefficients A036039. (End)

			1.6616879496335941212958189227499507499644186350250682081897111680...

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

Robert G. Wilson v, Table of n, a(n) for n = 1..1011
Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, Section 6.10.
Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
M. D. Hirschhorn, A note on Somos' quadratic recurrence constant, J. Number Theory 131 (2011), 2061-2063.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Cristinel Mortici, Estimating the Somos' quadratic recurrence constant, J. Number Theory 130 (2010), 2650-1657.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros 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.
Wikipedia, Somos' quadratic recurrence constant
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

Cf. A052129, A055209, A116603, A123851, A123852, A123853, A123854.
Cf. A114124 (log).
Cf. A036039, A090998, A135002, A188834, A259235, A274181.

RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2010 *)
Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Apr 07 2014, after Jonathan Sondow *)

{a(n) = if( n<-1, 0, n++; default( realprecision, n+2); floor( prodinf( k=1, k^2^-k)* 10^n) % 10)};

prodinf(n=1,n^2^-n) \\ Charles R Greathouse IV, Apr 07 2013

from mpmath import polylog, diff, exp, mp
mp.dps = 120
somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))
A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Nov 23 2024

Equals Product_{n>=1} n^(1/2^n). - Jonathan Sondow, Apr 07 2013
Equals exp(A114124) = A188834/2 = sqrt(A259235). - Hugo Pfoertner, Nov 23 2024
From Jwalin Bhatt, Apr 02 2025: (Start)
Equals exp(-PolyLog'(0,1/2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.
Equals Product_{n>=1} (1+1/n)^(1/2^n).
Equals exp(Sum_{n>=2} log(n)/2^n).
Equals 2*exp(Sum_{n>=1} (log(1+1/n)-1/n)/2^n). (End)

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

Original entry on oeis.org

1, 1, 2, 12, 576, 1658880, 16511297126400, 1908360529573854283038720000, 29134719286683212541013468732221146917416153907200000000

Offset: 0

Reinhard Zumkeller, Feb 12 2002

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 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
Unreduced denominators of: f(1) = 1, f(n) = f(n-1) + f(n-1)/(n-1). - Daniel Suteu, Jul 29 2016

			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 + ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Sung-Hyuk Cha, On the k-ary Tree Combinatorics.
Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 172 (2017), 145-159.
Chao-Ping Chen and X. F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, Volume 166, September 2016, Pages 31-40.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 16.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006.
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros 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.
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.

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

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 *)
a[n_] := Product[ k^(2^(n - k)), {k,1,n}] (* Jonathan Sondow, Mar 17 2014 *)
NestList[{#[[1]]+1,#[[1]]*#[[2]]^2}&,{1,1},10][[All,2]] (* Harvey P. Dale, Jul 30 2018 *)

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

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
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
A088679(n+1)/a(n) = n+1. -Daniel Suteu, Jul 29 2016

A116603 Coefficients in asymptotic expansion of sequence A052129.

Original entry on oeis.org

1, 2, -1, 4, -21, 138, -1091, 10088, -106918, 1279220, -17070418, 251560472, -4059954946, 71250808916, -1351381762990, 27552372478592, -601021307680207, 13969016314470386, -344653640328891233, 8997206549370634644, -247772400254700937149, 7178881153198162073002

Offset: 0

Michael Somos, Feb 18 2006

sign

			G.f. = 1 + 2*x - x^2 + 4*x^3 - 21*x^4 + 138*x^5 - 1091*x^6 + 10088*x^7 + ...

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

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 172 (2017), 145-159.
Chao-Ping Chen and X.-F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, 166 (2016), 31-40.
Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article no. 6.
Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.
Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:0610499 [math.CA], 2006.
Jonathan Sondow and Petros 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.
Eric Weisstein's World of Mathematics, Goebel's Sequence.
Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.
Aimin Xu, Approximations of the generalized-Euler-constant function and the generalized Somos' quadratic recurrence constant, Journal of Inequalities and Applications, Vol. 2019 (2019), Article No. 198.

Cf. A052129, A112302, A123851, A123852, A123853, A123854.

Cf. A000670, A084785.

terms = 20; A[] = 1; Do[A[x] = -A[x] + 2/A[x/(1+x)]^(-1/2)*(1+x) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 28 2011, updated Jan 12 2018 *)

{a(n) = my(A); if( n<0, 0, A=1; for( k=1, n, A = truncate( A + O(x^k)) + x * O(x^k); A = -A + 2 / subst(A^(-1/2), x, x/(1 + x)) * (1 + x);); polcoeff(A, n))};

a(0) = 1; thereafter, a(n) = (1/n)*Sum_{j=1..n} (-1)^(j-1)*2*b(j)*a(n-j), where b(j) = A000670(j) [Nemes]. - N. J. A. Sloane, Sep 11 2017
G.f. A(x) satisfies (1 + x)^2 = A(x)^2 / A(x/(1 + x)).
A003504(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where C = 1.04783144757... (see A115632).
A052129(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).
From Seiichi Manyama, May 26 2025: (Start)
G.f.: Product_{k>=1} (1 + k*x)^(1/2^k).
G.f.: exp(2 * Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A084785. (End)
a(n) ~ (-1)^(n+1) * (n-1)! / log(2)^(n+1). - Vaclav Kotesovec, May 27 2025

A123852 Decimal expansion of (1(2(3*...)^(1/3))^(1/3))^(1/3).

Original entry on oeis.org

1, 1, 5, 6, 3, 6, 2, 6, 8, 4, 3, 3, 2, 2, 6, 9, 7, 1, 6, 8, 5, 3, 3, 7, 0, 3, 2, 2, 8, 8, 7, 3, 6, 9, 3, 5, 6, 5, 1, 3, 0, 1, 4, 5, 4, 3, 8, 9, 1, 8, 8, 8, 6, 3, 7, 9, 9, 9, 2, 5, 9, 5, 9, 8, 9, 8, 3, 1, 7, 7, 8, 1, 6, 0, 7, 2, 8, 2, 6, 1, 9, 4, 6, 0, 7, 9, 0, 8, 1, 3, 3, 8, 2, 0, 3, 7, 8, 3, 1, 7

Offset: 1

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

Cubic recurrence constant (see A123851): a cubic analog of Somos's quadratic recurrence constant A112302.

			1.156362684332269716853370322887369356513014543891888637999259598983177816...

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

Kh. Hessami Pilehrood and T. Hessami Pilehrood, Vacca-type series for values of the generalized-Euler-constant function and its derivative, arXiv:0808.0410 [math.NT], 2008.
Kh. Hessami Pilehrood and T. Hessami Pilehrood, Vacca-type series for values of the generalized-Euler-constant function and its derivative, Journal of Integer Sequences 13 (2010), Article 10.7.3.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332(1) (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.

Cf. A052129, A112302, A116603, A123851, A123853, A123854.

Take[RealDigits[Product[N[n^3^-n,200], {n,400}]][[1]], 100]
RealDigits[Exp[-D[PolyLog[n, 1/3], n]/.n->0], 10, 100][[1]] (* Jean-François Alcover, Jan 28 2014 *)

prodinf(n=1, n^(1/3^n)) \\ Michel Marcus, Aug 03 2019

Equals Product_{n>=1} n^(1/3^n).

References updated by R. J. Mathar, Aug 12 2010

A164334 Quartic recurrence sequence a(0) = 1, a(n) = n*a(n-1)^4.

Original entry on oeis.org

1, 1, 2, 48, 21233664, 1016411962239204484414785454080

Offset: 0

David Willingham (D.Willingham(AT)wmin.ac.uk), Aug 13 2009

nonn

Number of different orderings for n-input trees in a Free Quaternary Decision Diagram.

The next term has 121 digits. - Harvey P. Dale, Dec 19 2016

G. C. Greubel, Table of n, a(n) for n = 0..7

Quartic extension of A052129 and A123851.

nxt[{n_,a_}]:={n+1,(n+1)a^4}; NestList[nxt,{0,1},5][[All,2]] (* Harvey P. Dale, Dec 19 2016 *)

a(n) = if (n==0, 1, n*a(n-1)^4); \\ Michel Marcus, Sep 14 2017

a(0) = 1, a(n) = n*a(n-1)^4.

A164335 Quintic recurrence sequence a(0) = 1, a(n) = n*a(n-1)^5.

Original entry on oeis.org

1, 1, 2, 96, 32614907904, 184523119031305377426211669050277696887837070322565120

Offset: 0

David Willingham (D.Willingham(AT)wmin.ac.uk), Aug 13 2009

nonn

Number of different orderings for n-input trees in a Free Quinary Decision Diagram.

a(7) onward have more than 1000 digits. - G. C. Greubel, Sep 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..6

Quintic extension of A052129, A123851 and A164334.

nxt[{n_, a_}] := {n + 1, (n + 1) a^5}; NestList[nxt, {0, 1}, 5][[All, 2]] (* G. C. Greubel, Sep 14 2017 *)

a(n) = if (n==0, 1, n*a(n-1)^5); \\ Michel Marcus, Sep 14 2017

a(0) = 1, a(n) = n*a(n-1)^5.

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A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

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A123853 Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.

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A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

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

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A116603 Coefficients in asymptotic expansion of sequence A052129.

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A123852 Decimal expansion of (1*(2*(3*...)^(1/3))^(1/3))^(1/3).

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A123852 Decimal expansion of (1(2(3*...)^(1/3))^(1/3))^(1/3).