A255433 -id:A255433 - 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

A272246 a(n) = Product_{k=0..n} (n^3 + k^3).

Original entry on oeis.org

0, 2, 1152, 1428840, 3488808960, 15044494500000, 105235903511101440, 1119277024472896248960, 17216259547948971039129600, 368066786222106315186876633600, 10591209807103301277597696000000000, 399472472359100444604916002033020774400

Offset: 0

Vaclav Kotesovec, Apr 23 2016

Cf. A126804, A272244, A272247, A272248, A367823, A367833.

Cf. A255433, A323541, A324426.

Table[Product[n^3+k^3,{k,0,n}],{n,0,12}]

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

A158621 Partial products of A001093.

Original entry on oeis.org

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

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... .

A375840 a(n) = Product_{k=0..n} (k^3 + n).

Original entry on oeis.org

0, 2, 60, 3960, 505920, 111945600, 39501498960, 20891200176000, 15785674348953600, 16407441209402496000, 22748452701706791576000, 41018285140626186366336000, 94161166261926730618189824000, 270252010494895412092926136320000, 954766647796042233397162343121696000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A073017, A255433.

Cf. A375839, A375841, A375842, A375843, A375844, A375845.

Table[Product[k^3 + n, {k, 0, n}], {n, 0, 15}]

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

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.

A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 8, 9, 1, 0, 216, 251, 36, 1, 0, 13824, 16280, 2555, 100, 1, 0, 1728000, 2048824, 335655, 15055, 225, 1, 0, 373248000, 444273984, 74550304, 3587535, 63655, 441, 1, 0, 128024064000, 152759224512, 26015028256, 1305074809, 25421200, 214918, 784, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
1,
0, 1,
0, 1,       1,
0, 8,       9,       1,
0, 216,     251,     36,     1,
0, 13824,   16280,   2555,   100,   1,
0, 1728000, 2048824, 335655, 15055, 225, 1.

Peter Luschny, The P-transform.

Variant: A249677.
Cf. A007318 (order 0), A132393 (order 1), A269944 (order 2).
Cf. A000442, A000537, A255433.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + (n-1)^3*T(n-1, k))) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, k_] := T[n, k] = Which[n == k, 1, k < 0 || k > n, 0, True, T[n - 1, k - 1] + (n - 1)^3 T[n - 1, k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

T(n,1) = ((n-1)!)^3 for n>=1 (cf. A000442).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
Row sums: Product_{k=1..n} ((k-1)^3+1) for n>=0 (cf. A255433).

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.

A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 18, 18, 1, 0, 504, 648, 72, 1, 0, 32760, 47160, 7200, 200, 1, 0, 4127760, 6305040, 1141560, 45000, 450, 1, 0, 895723920, 1416456720, 283704120, 13741560, 198450, 882, 1, 0, 308129028480, 498072032640, 106386981120, 5876519040, 106616160, 691488, 1568, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
[1]
[0, 1]
[0, 2,       1]
[0, 18,      18,      1]
[0, 504,     648,     72,      1]
[0, 32760,   47160,   7200,    200,   1]
[0, 4127760, 6305040, 1141560, 45000, 450, 1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A268434 (order 2).

Cf. A255433, A163102, A269947, A269948.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + ((n-1)^3+k^3) * T(n-1, k) )) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^3 + k^3)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

T(n,k) = Sum_{j=k..n} A269947(n,j)*A269948(j,k).
T(n,1) = Product_{k=1..n} (k-1)^3+1 for n>=1 (cf. A255433).
T(n,n-1) = (n-1)^2*n^2/2 for n>=1 (cf. A163102).

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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A272246 a(n) = Product_{k=0..n} (n^3 + k^3).

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A158621 Partial products of A001093.

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

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A375840 a(n) = Product_{k=0..n} (k^3 + n).

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

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A269947 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.

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