A255435 -id:A255435 - cp's OEIS frontend

A101686 a(n) = Product_{i=1..n} (i^2 + 1).

1, 2, 10, 100, 1700, 44200, 1635400, 81770000, 5315050000, 435834100000, 44019244100000, 5370347780200000, 778700428129000000, 132379072781930000000, 26078677338040210000000, 5893781078397087460000000, 1514701737148051477220000000

Offset: 0

Ralf Stephan, Dec 13 2004

nonn

Sum of all coefficients in Product_{k=0..n} (x + k^2).
Row sums of triangle of central factorial numbers (A008955).
"HANOWA" is a matrix whose eigenvalues lie on a vertical line. It is an N X N matrix with 2 X 2 blocks with identity matrices in the upper left and lower right blocks and diagonal matrices containing the first N integers in the upper right and lower left blocks. In MATLAB, the following code generates the sequence... for n=0:2:TERMS*2 det(gallery('hanowa',n)) end. - Paul Max Payton, Mar 31 2005
Cilleruelo shows that a(n) is a square only for n = 0 and 3. - Charles R Greathouse IV, Aug 27 2008
a(n) = A231530(n)^2 + A231531(n)^2. - Stanislav Sykora, Nov 10 2013

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 559-561, Section 147. - N. J. A. Sloane, May 29 2014

Stanislav Sykora, Table of n, a(n) for n = 0..252
Javier Cilleruelo, Squares in (1^2+1)...(n^2+1), Journal of Number Theory 128:8 (2008), pp. 2488-2491.
Thang Pang Ern, Finding Squares in a Product of Squares, arXiv:2411.00012 [math.NT], 2024.
Erhan Gürela and 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.
V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012.

Equals 2 * A051893(n+1), n>0. Cf. A156648.
Cf. A255433, A255434, A255435, A003703, A009454.
Cf. A105750, A105751.

p := n -> mul(x^2+1, x=0..n):
seq(p(i), i=0..14); # Gary Detlefs, Jun 03 2010

Table[Product[k^2+1,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 11 2013 *)
Table[Pochhammer[I, n + 1] Pochhammer[-I, n + 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)
Table[Abs[Pochhammer[1 + I, n]]^2, {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2016 *)

a(n)=prod(k=1,n,k^2+1) \\ Charles R Greathouse IV, Aug 27 2008

{a(n)=if(n==0, 1, 1-polcoeff(sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, 1+j^2*x+x*O(x^n))), n))} \\ Paul D. Hanna, Jan 07 2013

from math import prod
def A101686(n): return prod(i**2+1 for i in range(1,n+1)) # Chai Wah Wu, Feb 22 2024

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^2*x). - Paul D. Hanna, Jan 07 2013
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((k+1)^2+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) ~ (n!)^2 * sinh(Pi)/Pi. - Vaclav Kotesovec, Nov 11 2013
From Vladimir Reshetnikov, Oct 25 2015: (Start)
a(n) = Gamma(n+1+i)*Gamma(n+1-i)*sinh(Pi)/Pi.
a(n) ~ 2*exp(-2*n)*n^(2*n+1)*sinh(Pi).
G.f. for 1/a(n): hypergeom([1], [1-i, 1+i], x).
E.g.f. for a(n)/n!: hypergeom([1-i, 1+i], [1], x), where i=sqrt(-1).
D-finite with recurrence: a(0) = 1, a(n) = (n^2+1)*a(n-1). (End)
a(n+3)/a(n+2) - 2 a(n+2)/a(n+1) + a(n+1)/a(n) = 2. - Robert Israel, Oct 25 2015
a(n) = A003703(n+1)^2 + A009454(n+1)^2. - Vladimir Reshetnikov, Oct 15 2016
a(n) = A105750(n)^2 + A105751(n)^2. - Ridouane Oudra, Dec 15 2021

More terms from Charles R Greathouse IV, Aug 27 2008
Simpler definition from Gary Detlefs, Jun 03 2010
Entry revised by N. J. A. Sloane, Dec 22 2012
Minor edits by Vaclav Kotesovec, Mar 13 2015

A255434 Product_{k=0..n} (k^4+1).

Original entry on oeis.org

1, 2, 34, 2788, 716516, 448539016, 581755103752, 1397375759212304, 5725048485492809488, 37567768161803815860256, 375715249386199962418420256, 5501222681512739849730509388352, 114078854746529686263861573186255424, 3258320249270380899068414253345827420288

Offset: 0

Vaclav Kotesovec, Feb 23 2015

Harvey P. Dale, Table of n, a(n) for n = 0..144
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.

Cf. A002523, A101686, A255433, A255435, A144663.

Table[Product[k^4 + 1, {k, 0, n}], {n, 0, 15}]
FoldList[Times,Range[0,15]^4+1] (* Harvey P. Dale, Nov 01 2022 *)

a(n) = prod(k=1, n, 1+k^4); \\ Michel Marcus, Jan 25 2016

a(n) ~ 2 * (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)) * n^(4*n+2) / exp(4*n).

a(n) ~ A258870 * (n!)^4. - Vaclav Kotesovec, May 16 2022

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

A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

Original entry on oeis.org

0, 2, 67584, 7924375800, 2876035930521600, 2693451205324687500000, 5648896640332217707816550400, 23819277009290664033936067933468800, 185754160490281505676324140907107450880000, 2507604631016507710974687639612411760216253760000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

In general, for p>=1, Product_{k=0..n} (n^p + k^p) ~ sqrt(2) * n^(p*(n+1)) * exp(n*Sum_{j>=1} (-1)^(j+1) / (j*(1 + j*p))).

Cf. A126804, A272244, A272246, A272247, A367823, A367833.

Cf. A255435, A323543, A324438.

Table[Product[n^5+k^5,{k,0,n}],{n,0,10}]

a(n) ~ 2^(2*n+1/2) * phi^(sqrt(5)*n) * n^(5*n+5) / exp((5-sqrt(phi)*Pi/5^(1/4))*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A375842 a(n) = Product_{k=0..n} (k^5 + n).

Original entry on oeis.org

0, 2, 204, 103320, 182819520, 886613565600, 9973411835945040, 230723130088707984000, 10026790739932043582668800, 762878670971305314645055065600, 96049580303627434572710125880376000, 19110279123993980912852049573590159155200

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A307216, A255435.

Cf. A375839, A375840, A375841, A375843, A375844, A375845.

Table[Product[k^5 + n, {k, 0, n}], {n, 0, 15}]

a(n) ~ n^(5*n + 3) / exp(5*n - Pi * sqrt(2*(1+sqrt(5))) * n^(1/5) / 5^(1/4)).

A144664 Decimal expansion of Product_{n>=2} (n^5-1)/(n^5+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.92878693579955245375146991...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986, A255435.

p = 2*Gamma[2-(-1)^(1/5)] * Gamma[2+(-1)^(2/5)] * Gamma[2-(-1)^(3/5)] * Gamma[2+(-1)^(4/5)] / (Gamma[2+(-1)^(1/5)] * Gamma[2-(-1)^(2/5)] * Gamma[2+(-1)^(3/5)] * Gamma[2-(-1)^(4/5)]); RealDigits[Re[p], 10, 110][[1]] (* Jean-François Alcover, Feb 11 2013, updated Nov 18 2015 *)
RealDigits[Re[N[Product[(n^5 - 1)/(n^5 + 1), {n, 2, Infinity}], 110]]][[1]] (* Bruno Berselli, Apr 02 2013 *)

More terms from Jean-François Alcover, Feb 11 2013

cp's OEIS Frontend

A101686 a(n) = Product_{i=1..n} (i^2 + 1).

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A255434 Product_{k=0..n} (k^4+1).

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

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A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

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A375842 a(n) = Product_{k=0..n} (k^5 + n).

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A144664 Decimal expansion of Product_{n>=2} (n^5-1)/(n^5+1).

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