A158621 -id:A158621 - cp's OEIS frontend

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

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

A158623 Denominator of the reduced fraction A158620(n)/A158621(n).

Original entry on oeis.org

9, 18, 10, 45, 63, 28, 108, 135, 55, 198, 234, 91, 315, 360, 136, 459, 513, 190, 630, 693, 253, 828, 900, 325, 1053, 1134, 406, 1305, 1395, 496, 1584, 1683, 595, 1890, 1998, 703, 2223, 2340, 820, 2583, 2709, 946, 2970, 3105, 1081, 3384, 3528, 1225, 3825

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, ...

			a(2) = 9 = denominator of (2^3-1)/2^3+1 = 7/9. a(3) = 18 = denominator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18. a(4) = 10 = denominator 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) = 45 = denominator 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.

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

A158623 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; denom(%) ; end: seq(A158623(n),n=2..100) ; # R. J. Mathar, Mar 27 2009

Denominator of (Product_{k=2..n} (k^3-1)) / Product_{k=2..n} (k^3+1) = denominator of Product_{k=2..n} A068601(k)/A001093(k).
A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). Conjecture: a(n) = 3a(n-3) - 3a(n-6) + a(n-9), so trisections are A152996, A060544 and 3*A081266. - R. J. Mathar, Mar 27 2009
Empirical g.f.: -x^2*(x^8 - 2*x^5 + 9*x^4 + 18*x^3 + 10*x^2 + 18*x + 9) / ((x-1)^3*(x^2 + x + 1)^3). - Colin Barker, May 09 2013

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

A255433 a(n) = Product_{k=0..n} (k^3+1).

Original entry on oeis.org

1, 2, 18, 504, 32760, 4127760, 895723920, 308129028480, 158070191610240, 115391239875475200, 115506631115350675200, 153854832645647099366400, 266015005644323834804505600, 584700982406223788900303308800, 1605004196705084300531332582656000

Offset: 0

Vaclav Kotesovec, Feb 23 2015

G. C. Greubel, Table of n, a(n) for n = 0..150
Erhan Gürela, Ali Ulas Özgür Kisisel, A note on the products (1^mu + 1)(2^mu + 1)···(n^mu + 1), Journal of Number Theory, Volume 130, Issue 1, January 2010, Pages 187-191.
Chuan Ze Niu, (1^3+1)(2^3+1)...(n^3+1) is not a cube, arXiv:1612.08158 [math.NT], 2016.

Cf. A101686, A158621, A255434, A255435.

Table[Product[k^3 + 1, {k, 0, n}], {n, 0, 20}]
FullSimplify[Table[(Cosh[Sqrt[3]*Pi/2] * Gamma[2+n] * Gamma[1/2 - I*Sqrt[3]/2 + n] * Gamma[1/2 + I*Sqrt[3]/2 + n])/Pi, {n, 0, 20}]]
FoldList[Times,Range[0,20]^3+1] (* Harvey P. Dale, Jul 07 2017 *)

a(n) = prod(k=1, n, 1+k^3); \\ Michel Marcus, Jan 25 2016

a(n) ~ 2*sqrt(2*Pi) * cosh(sqrt(3)*Pi/2) * n^(3*n + 3/2) / exp(3*n).

a(n) = 2*A158621(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

A256019 a(n) = Sum_{i=1..n-1} (i^3 * a(i)), a(1)=1.

Original entry on oeis.org

1, 1, 9, 252, 16380, 2063880, 447861960, 154064514240, 79035095805120, 57695619937737600, 57753315557675337600, 76927416322823549683200, 133007502822161917402252800, 292350491203111894450151654400, 802502098352542150265666291328000

Offset: 1

Vaclav Kotesovec, Mar 13 2015

nonn

a(n) = A158621(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Cf. A001710, A051893, A158621, A256020.

Clear[a]; a[1]=1; a[n_]:= a[n] = Sum[i^3*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}]
Flatten[{1, Table[FullSimplify[Cosh[Sqrt[3]*Pi/2] * Gamma[n+1] * Gamma[n-1/2 - I*Sqrt[3]/2] * Gamma[n-1/2 + I*Sqrt[3]/2] / (2*Pi)],{n,2,20}]}]

Product_{i=2..n-1} (i^3 + 1), for n>2.
a(n) ~ cosh(sqrt(3)*Pi/2) / (2*Pi) * ((n-1)!)^3.
a(n) = A255433(n-1)/2.

cp's OEIS Frontend

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

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A158623 Denominator of the reduced fraction A158620(n)/A158621(n).

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A255433 a(n) = Product_{k=0..n} (k^3+1).

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A158620 Partial products of A068601.

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A256019 a(n) = Sum_{i=1..n-1} (i^3 * a(i)), a(1)=1.

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