A158623 -id:A158623 - cp's OEIS frontend

A158621 Partial products of A001093.

9, 252, 16380, 2063880, 447861960, 154064514240, 79035095805120, 57695619937737600, 57753315557675337600, 76927416322823549683200, 133007502822161917402252800, 292350491203111894450151654400

Offset: 2

Jonathan Vos Post, Mar 23 2009

A158620(n) = PRODUCT[k=2..n](k^3-1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n).

			a(2) = 2^3+1 = 9. a(3) = (2^3+1)*(3^3+1) = 9 * 28 = 252. a(4) = (2^3+1)*(3^3+1)*(4^3+1) = 9 * 28 * 65 = 16380.

Cf. A001093, A016921, A068601, A158620, A158622-A158624, A255433.

Table[Product[(k^3+1),{k,2,n}],{n,2,20}] (* Vaclav Kotesovec, Jul 11 2015 *)
FoldList[Times,Range[2,20]^3+1] (* Harvey P. Dale, Oct 15 2017 *)

PRODUCT[k=2..n](k^3+1) = PRODUCT[k=2..n]A001093(k).

a(n) ~ sqrt(2*Pi) * cosh(sqrt(3)*Pi/2) * n^(3*n+3/2) / exp(3*n). - Vaclav Kotesovec, Jul 11 2015

A158620 Partial products of A068601.

Original entry on oeis.org

7, 182, 11466, 1421784, 305683560, 104543777520, 53421870312720, 38891121587660160, 38852230466072499840, 51673466519876424787200, 89240076679826585607494400, 195971208388899181994057702400

Offset: 2

Jonathan Vos Post, Mar 23 2009

A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n).

Also the determinant of the n X n matrix given by m(i,j) = i^3 if i=j and 1 otherwise. For example, Det[{{1,1,1, 1},{1,8,1,1},{1,1,27,1},{1,1,1,64}}] = 11466 = a(4). - John M. Campbell, May 20 2011

			a(2) = 2^3-1 = 7.
a(3) = (2^3-1)*(3^3-1) = 7 * 26 = 182.
a(4) = (2^3-1)*(3^3-1)*(4^3-1) = 7 * 26 * 63 = 11466.

G. C. Greubel, Table of n, a(n) for n = 2..180

Cf. A000217, A005448, A016921, A068601, A158621-A158624, A010791.

Rest[FoldList[Times,1,Range[2,15]^3-1]] (* Harvey P. Dale, Apr 18 2015 *)

a(n) = prod(k = 2, n, k^3 - 1); \\ Michel Marcus, Sep 29 2013

Product_{k=2..n} (k^3-1) = Product_{k=2..n} A068601(k).

a(n) ~ 2^(3/2) * sqrt(Pi) * cosh(sqrt(3)*Pi/2) * n^(3*n+3/2) / (3 * exp(3*n)). - Vaclav Kotesovec, Jul 11 2015

A158622 Numerator of the reduced fraction A158620(n)/A158621(n).

Original entry on oeis.org

7, 13, 7, 31, 43, 19, 73, 91, 37, 133, 157, 61, 211, 241, 91, 307, 343, 127, 421, 463, 169, 553, 601, 217, 703, 757, 271, 871, 931, 331, 1057, 1123, 397, 1261, 1333, 469, 1483, 1561, 547, 1723, 1807, 631, 1981, 2071, 721, 2257, 2353, 817, 2551, 2653, 919, 2863

Offset: 2

Jonathan Vos Post, Mar 23 2009

A158620(n) = Product_{k=2..n} (k^3-1). A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n). The reduced fractions are 7/9, 13/18, 7/10, 31/45, 43/63, 19/28, 73/108, 91/135, 37/55, 133/198, ...
Is this the same as A046163? - R. J. Mathar, Mar 27 2009
Apparently a(n) = A130770(n) for 2 <= n <= 53. - Georg Fischer, Oct 24 2018

			a(2) = 7 = numerator of (2^3-1)/2^3+1 = 7/9.
a(3) = 13 = numerator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18.
a(4) = 7 = = numerator of ((2^3-1)*(3^3-1)*(4^3-1))/((2^3+1)*(3^3+1)*(4^3+1)) = (7 * 26 * 63)/(9 * 28 * 65) = 11466/16380 = 7/10.
a(5) = 31 = numerator of ((2^3-1)(3^3-1)(4^3-1)(5^3-1))/((2^3+1)(3^3+1)(4^3+1)(5^3+1)) = 1421784/2063880 = 31/45.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A001093, A016921, A068601, A130770, A158620-A158621, A158623.

A158622 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; numer(%) ; end: seq(A158622(n),n=2..100) ; # R. J. Mathar, Mar 27 2009

Table[Product[k^3-1,{k,2,n}]/Product[k^3+1,{k,2,n}],{n,2,60}]//Numerator (* Harvey P. Dale, Feb 26 2020 *)

Numerator of (Product_{k=2..n} (k^3-1))/Product_{k=2..n} (k^3+1) = numerator of Product_{k=2..n} A068601(k)/A001093(k).
A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). - R. J. Mathar, Mar 27 2009
Empirical g.f.: -x^2*(x^8 + x^7 + x^6 - 2*x^5 + 4*x^4 + 10*x^3 + 7*x^2 + 13*x + 7) / ((x-1)^3*(x^2 + x + 1)^3). - Colin Barker, May 09 2013

More terms from R. J. Mathar, Mar 27 2009

cp's OEIS Frontend

A158621 Partial products of A001093.

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A158620 Partial products of A068601.

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A158622 Numerator of the reduced fraction A158620(n)/A158621(n).

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