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%I #83 Feb 22 2024 13:33:29
%S 1,2,10,100,1700,44200,1635400,81770000,5315050000,435834100000,
%T 44019244100000,5370347780200000,778700428129000000,
%U 132379072781930000000,26078677338040210000000,5893781078397087460000000,1514701737148051477220000000
%N a(n) = Product_{i=1..n} (i^2 + 1).
%C Sum of all coefficients in Product_{k=0..n} (x + k^2).
%C Row sums of triangle of central factorial numbers (A008955).
%C "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
%C Cilleruelo shows that a(n) is a square only for n = 0 and 3. - _Charles R Greathouse IV_, Aug 27 2008
%C a(n) = A231530(n)^2 + A231531(n)^2. - _Stanislav Sykora_, Nov 10 2013
%D 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
%H Stanislav Sykora, <a href="/A101686/b101686.txt">Table of n, a(n) for n = 0..252</a>
%H Javier Cilleruelo, <a href="http://dx.doi.org/10.1016/j.jnt.2007.11.001">Squares in (1^2+1)...(n^2+1)</a>, Journal of Number Theory 128:8 (2008), pp. 2488-2491.
%H Erhan Gürela and Ali Ulas Özgür Kisisel, <a href="http://dx.doi.org/10.1016/j.jnt.2009.07.014">A note on the products (1^mu + 1)(2^mu + 1)···(n^mu + 1)</a>, Journal of Number Theory, Volume 130, Issue 1, January 2010, Pages 187-191.
%H V. H. Moll, <a href="http://www.tulane.edu/~vhm/papers_html/xn-final.pdf">An arithmetic conjecture on a sequence of arctangent sums</a>, 2012.
%F 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
%F 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
%F a(n) ~ (n!)^2 * sinh(Pi)/Pi. - _Vaclav Kotesovec_, Nov 11 2013
%F From _Vladimir Reshetnikov_, Oct 25 2015: (Start)
%F a(n) = Gamma(n+1+i)*Gamma(n+1-i)*sinh(Pi)/Pi.
%F a(n) ~ 2*exp(-2*n)*n^(2*n+1)*sinh(Pi).
%F G.f. for 1/a(n): hypergeom([1], [1-i, 1+i], x).
%F E.g.f. for a(n)/n!: hypergeom([1-i, 1+i], [1], x), where i=sqrt(-1).
%F D-finite with recurrence: a(0) = 1, a(n) = (n^2+1)*a(n-1). (End)
%F a(n+3)/a(n+2) - 2 a(n+2)/a(n+1) + a(n+1)/a(n) = 2. - _Robert Israel_, Oct 25 2015
%F a(n) = A003703(n+1)^2 + A009454(n+1)^2. - _Vladimir Reshetnikov_, Oct 15 2016
%F a(n) = A105750(n)^2 + A105751(n)^2. - _Ridouane Oudra_, Dec 15 2021
%p p := n -> mul(x^2+1, x=0..n):
%p seq(p(i), i=0..14); # _Gary Detlefs_, Jun 03 2010
%t Table[Product[k^2+1,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 11 2013 *)
%t Table[Pochhammer[I, n + 1] Pochhammer[-I, n + 1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 25 2015 *)
%t Table[Abs[Pochhammer[1 + I, n]]^2, {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 16 2016 *)
%o (PARI) a(n)=prod(k=1,n,k^2+1) \\ _Charles R Greathouse IV_, Aug 27 2008
%o (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
%o (Python)
%o from math import prod
%o def A101686(n): return prod(i**2+1 for i in range(1,n+1)) # _Chai Wah Wu_, Feb 22 2024
%Y Equals 2 * A051893(n+1), n>0. Cf. A156648.
%Y Cf. A255433, A255434, A255435, A003703, A009454.
%Y Cf. A105750, A105751.
%K nonn
%O 0,2
%A _Ralf Stephan_, Dec 13 2004
%E More terms from _Charles R Greathouse IV_, Aug 27 2008
%E Simpler definition from _Gary Detlefs_, Jun 03 2010
%E Entry revised by _N. J. A. Sloane_, Dec 22 2012
%E Minor edits by _Vaclav Kotesovec_, Mar 13 2015
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