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A123851   A cubic recurrence: a(0) = 1, a(n) = n*a(n-1)^3. 9
1, 1, 2, 24, 55296, 845378412871680, 3624972460853492659595005581182702601633792000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A cubic analog of Somos's quadratic recurrence sequence A052129.

Terms a(7) onward are too big to include in data section. - G. C. Greubel, Aug 10 2019

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..8

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

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:0610499 [math.CA], 2006.

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.

Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory (2019), to appear.

FORMULA

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

EXAMPLE

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.

G.f. = 1 + x + 2*x^2 + 24*x^3 + 55296*x^4 + 845378412871680*x^5 + ...

MATHEMATICA

a[n_]:= If[n==0, 1, n*a[n-1]^3]; Table[a[n], {n, 0, 7}]

nxt[{n_, a_}]:={n+1, (n+1)a^3}; NestList[nxt, {0, 1}, 7][[All, 2]] (* Harvey P. Dale, May 25 2019 *)

PROG

(PARI) {a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^3^k))}; /* Michael Somos, Aug 07 2016 */

(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

(Sage) [1]+[prod((n-k)^(3^k) for k in (0..n-1)) for n in (1..8)] # G. C. Greubel, Aug 10 2019

(GAP) List([0..8], n-> Product([0..n-1], k-> (n-k)^(3^k)) ); # G. C. Greubel, Aug 10 2019

CROSSREFS

Cf. A052129, A112302, A116603, A123852, A123853, A123854.

Sequence in context: A108349 A000722 A098679 * A258824 A120122 A068943

Adjacent sequences:  A123848 A123849 A123850 * A123852 A123853 A123854

KEYWORD

easy,nonn

AUTHOR

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

EXTENSIONS

Corrected by Harvey P. Dale, May 25 2019

STATUS

approved

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)