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

Author

Ralf Stephan, Dec 13 2004

Keywords

• nonn

Comments

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

References

• 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

Links

• 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.

Crossrefs

Equals 2 * A051893(n+1), n>0. Cf. A156648.

Cf. A255433, A255434, A255435, A003703, A009454.

Cf. A105750, A105751.

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

{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

Python

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

Formula

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

Extensions

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