A255434 -id:A255434 - 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

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

A272247 a(n) = Product_{k=0..n} (n^4 + k^4).

Original entry on oeis.org

0, 2, 8704, 104372388, 3087748038656, 194985808028125000, 23467500289618753093632, 4938279594477505466022892304, 1699491802948673617630927695904768, 907214902722906584628050661111614570016, 719684491044699824651608981274624000000000000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

Cf. A126804, A272244, A272246, A272248, A367823, A367833.

Cf. A255434, A323542.

Table[Product[n^4+k^4,{k,0,n}],{n,0,10}]

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

A255435 Product_{k=0..n} (k^5 + 1).

Original entry on oeis.org

1, 2, 66, 16104, 16506600, 51599631600, 401290334953200, 6744887949893385600, 221023233230056352726400, 13051421922234827628493920000, 1305155243645404997677020493920000, 210197862299579765685879504586803840000, 52304164669591331834914454764848159918720000

Offset: 0

Vaclav Kotesovec, Feb 23 2015

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. A101686, A255433, A255434, A144664.

Table[Product[k^5 + 1, {k, 0, n}], {n, 0, 15}]

a(n) = prod(k=1, n, 1+k^5); \\ Michel Marcus, Jan 25 2016

a(n) ~ c * n^(5*n+5/2) / exp(5*n), where c = 4 * sqrt(2) * Pi^(5/2) / abs(Gamma(3/4 + sqrt(5)/4 + i*sqrt(5/8 + sqrt(5)/8)) * Gamma(3/4 - sqrt(5)/4 + i*sqrt(5/8 - sqrt(5)/8)))^2 = 205.260576646551853296... .

A375841 a(n) = Product_{k=0..n} (k^4 + n).

Original entry on oeis.org

0, 2, 108, 19152, 8840000, 8908817400, 17303456226672, 59111538137501696, 331331804053754904576, 2885800103371503562500000, 37384163240259410286768056000, 694933775143924511454539020849152, 17989643936954432911290280974476623872, 632268529759009258574304284235050340614528

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A258870, A255434.

Cf. A375839, A375840, A375842, A375843, A375844, A375845.

Table[Product[k^4 + n, {k, 0, n}], {n, 0, 15}]

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

A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.8480540493529003921296501834...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
László Tóth, Transcendental Infinite Products Associated with the ±1 Thue-Morse Sequence, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.2.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A090986, A255434.

Digits := 120 :
m := 1:
for r from 2 to 10 do
omega := cos(Pi/r)+I*sin(Pi/r) :
x := (-1)^(m+1)*2*m*m!/r*mul( GAMMA(-m*omega^j)^(-(-1)^j),j=1..2*r-1) ;
x := Re(evalf(x)) ;
print(r,x) ;
od:

RealDigits[ -1/2*Pi*Csc[(-1)^(1/4)*Pi]*Csc[(-1)^(3/4)*Pi]*Sinh[Pi] // Re, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)
RealDigits[Re[N[Product[(n^4 - 1)/(n^4 + 1), {n, 2, Infinity}], 110]]][[1]] (* Bruno Berselli, Apr 02 2013 *)

Pi*sinh(Pi)/(cosh(Pi*sqrt(2))-cos(Pi*sqrt(2))) \\ Michel Marcus, Sep 07 2020

Equals Pi*sinh(Pi) / (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)). - Vaclav Kotesovec, Dec 08 2015

More terms from Jean-François Alcover, Feb 11 2013

A256020 a(n) = Sum_{i=1..n-1} (i^4 * a(i)), a(1)=1.

Original entry on oeis.org

1, 1, 17, 1394, 358258, 224269508, 290877551876, 698687879606152, 2862524242746404744, 18783884080901907930128, 187857624693099981209210128, 2750611340756369924865254694176, 57039427373264843131930786593127712, 1629160124635190449534207126672913710144

Offset: 1

Vaclav Kotesovec, Mar 13 2015

nonn

Cf. A001710, A051893, A255434, A256019.

Clear[a]; a[1]=1; a[n_]:= a[n] = Sum[i^4*a[i],{i,1,n-1}]; Table[a[n],{n,1,15}]
Flatten[{1,1, Table[Product[(i^4 + 1), {i,2,n-1}],{n,3,15}]}]
Join[{1},FoldList[Times,Range[15]^4+1]/2] (* Harvey P. Dale, Jul 29 2018 *)

a(n) = Product_{i=2..n-1} (i^4 + 1), for n>2.
a(n) ~ (cosh(Pi/sqrt(2))^2 * sin(Pi/sqrt(2))^2 + cos(Pi/sqrt(2))^2 * sinh(Pi/sqrt(2))^2) / (2*Pi^2) * ((n-1)!)^4.
a(n) = A255434(n-1)/2.

A354051 Decimal expansion of Sum_{k>=0} 1 / (k^4 + 1).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 16 2022, following a suggestion from Bernard Schott

Apart from leading digits the same as A256920. - R. J. Mathar, May 20 2022

			1.578477579667136838318022193245719235046672217327291327587486645793808...

Cf. A256920, A113319, A354052, A354053.

Cf. A002523, A255434, A354004.

evalf(1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))), 105);

RealDigits[Chop[N[Sum[1/(k^4 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^4 + 1))
```

Equals 1/2 + (Pi*(sinh(sqrt(2)*Pi) + sin(sqrt(2)*Pi))) / (2*sqrt(2)*(cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi))).

A274306 a(n) = Product_{k=1..n} (4*k^4+1).

Original entry on oeis.org

1, 5, 325, 105625, 108265625, 270772328125, 1403954521328125, 13484983177356640625, 220951449360988556640625, 5798870788479144669033203125, 231960630409954265905997158203125, 13584774319958971582784723570166015625

Offset: 0

N. J. A. Sloane, Jun 20 2016

nonn

Erhan Gürel, On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products, The American Mathematical Monthly 123.6 (2016): 597-599.

For squares in this sequence see A274307.

Cf. A255434.

Table[Product[4*k^4+1, {k,1,n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 10 2016 *)

a(n) = prod(k=1, n, 4*k^4+1); \\ Michel Marcus, Oct 10 2016

a(n) ~ (1 + cosh(Pi)) * 2^(2*n + 2) * n^(4*n + 2) / exp(4*n). - Vaclav Kotesovec, Oct 10 2016

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

Original entry on oeis.org

1, 2, 130, 94900, 388805300, 6075471617800, 283463279271694600, 33349454806314869690000, 8742392830201411514885050000, 4646074730467898538293540742100000, 4646079376542629006192079035640742100000, 8230817672466612927467651920537784356160200000

Offset: 0

Vaclav Kotesovec, May 16 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..103

Cf. A002604, A101686, A255433, A255434, A258871, A354052.

A354054 := proc(n)
    mul( k^6+1,k=0..n) ;
end proc:
seq(A354054(n),n=0..40) ; # R. J. Mathar, Jul 17 2023

Table[Product[k^6 + 1, {k, 0, n}], {n, 0, 15}]

PARI
```
a(n) = prod(k=1, n, k^6+1);
```

a(n) ~ (2*sinh(2*Pi) - 4*sinh(Pi)*cos(sqrt(3)*Pi)) * n^(6*n + 3) / exp(6*n).

a(n) ~ A258871 * (n!)^6.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A272247 a(n) = Product_{k=0..n} (n^4 + k^4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A255435 Product_{k=0..n} (k^5 + 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375841 a(n) = Product_{k=0..n} (k^4 + n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A256020 a(n) = Sum_{i=1..n-1} (i^4 * a(i)), a(1)=1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A354051 Decimal expansion of Sum_{k>=0} 1 / (k^4 + 1).

Views

Author

Keywords

Comments

Examples