A105751 -id:A105751 - 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

A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).

Original entry on oeis.org

1, 1, -1, -10, -10, 190, 730, -6620, -55900, 365300, 5864300, -28269800, -839594600, 2691559000, 159300557000, -238131478000, -38894192662000, -15194495654000, 11911522255750000, 29697351895900000, -4477959179352100000, -21683886333440500000, 2029107997508660900000

Offset: 0

Paul Barry, Apr 18 2005

Define u(n) as in A220448 and set f(n) = u(n)*f(n-1) for n >= 2, with f(1)=1 (this defines A220449). Then a(0)=1; a(n) = (-1)^(n+1)*f(n) for n >= 1. - N. J. A. Sloane, Dec 22 2012
From Peter Bala, Jun 03 2023: (Start)
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes.
Type 1: primes p that do not divide any element of the sequence {a(n)}. The first few type 1 primes appear to be {3, 7, 11, 23, 31, 47, 59}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. The first few type 2 primes appear to be {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97}.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002144.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined).
We conjecture that the sets of type 1 and type 3 primes taken together consist of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q. See A002145. (End)

N. J. A. Sloane, Table of n, a(n) for n = 0..100
V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A003703, A009454, A105751, A220448, A220449, A363409 - A363416.

A105750 := proc(n)
    mul(1-k*I,k=0..n) ;
    Re(%) ;
end proc: # R. J. Mathar, Jan 04 2013

x[n_] := x[n] = If[n == 1, 1, (x[n-1]+n)/(1-n*x[n-1])];
u[n_] := n*x[n-1]-1;
f[n_] := f[n] = If[n == 1, 1, u[n]*f[n-1]];
a[n_] := If[n == 0, 1, (-1)^(n+1)*f[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 17 2023, after N. J. A. Sloane *)

from sympy.functions.combinatorial.numbers import stirling
def A105750(n): return sum(stirling(n+1,n+1-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n+1>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = Re( Product_{k = 0..n} (1 - k*i) ).
Conjecture: a(n) -3*a(n-1) +(n^2-n+3)*a(n-2) +(-n^2+4*n-5)*a(n-3)=0. - R. J. Mathar, May 23 2014
From Peter Bala, May 28 2023: (Start)
a(n) = Sum_{k = 0..floor((n + 1)/2)} (-1)^k*|Stirling1(n+1, n-2*k+1)|, where Stirling1(n, k) = A048994(n,k).
(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(n^2 - 2*n + 2)*a(n-2) with a(0) = a(1) = 1 (see Moll, equation 1.16). Mathar's third-order recurrence above follows easily from this.
a(2*n) = (-1)^n*A009454(2*n+1) for n >= 0.
a(2*n-1) = (-1)^n*A003703(2*n) for n >= 1. (End)

Corrected by N. J. A. Sloane, Nov 05 2005

A231531 Imaginary part of Product_{k = 1..n} (k + i), i = sqrt(-1).

Original entry on oeis.org

0, 1, 3, 10, 40, 190, 1050, 6620, 46800, 365300, 3103100, 28269800, 271627200, 2691559000, 26495469000, 238131478000, 1394099824000, -15194495654000, -936096296850000, -29697351895900000, -819329864480400000, -21683886333440500000, -570263312237604700000, -15145164178973569000000, -409583160925827252000000

Offset: 0

Stanislav Sykora, Nov 10 2013

Extension of factorial(n) to factim(n,m) defined by the recurrence a(0)=1, a(n)=a(n-1)*(n+m*i). Hence n! = factim(n,0), while the current sequence shows the imaginary parts of factim(n,1). The real parts are in A231530 and squares of magnitudes are in A101686.
From Peter Bala, Jun 01 2023: (Start)
Compare with A105751(n) = the imaginary part of Product_{k = 0..n} (1 + k*sqrt(-1)). Moll (2012) studied the prime divisors of the terms of A105750 -  the real part of Product_{k = 0..n} (1 + k*sqrt(-1)) - and divided the primes into three types. Calculation suggests that a similar division holds in this case.
Type 1: the prime p does not divide any element of the sequence. It appears that for this sequence, unlike in A105750, there are no type 1 primes; i.e., every prime p divides some term of the sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of p = 2 and all primes of the form p == 1 (mod 4). See A002144.
Moll's conjecture 5.5 about type 2 primes extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
(ii) for a type 2 prime p, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes is A002145, primes of the form 4*k + 3. (End)

			factim(5,1) = -90+190*i. Hence a(5) = 190.
From _Peter Bala_, Jun 01 2023: (Start)
Asymptotic linearity for the type 2 prime p = 5: the sequence of 5-adic valuations [ v_5(a(n)) : n = 1..100] = [0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 6, 5, 6, 6, 7, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 11, 11, 11, 13, 11, 12, 12, 13, 12, 12, 13, 13, 13, 14, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 18, 19, 18, 18, 18, 19, 19, 19, 20, 19, 20, 21, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 25, 24, 24, 24, 25, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in line with Moll's conjecture 5.5 above.
Oscillatory behavior for the type 3 prime p = 3: the sequence of 3-adic valuations [ v_3(a(n)) : n = 1..100] = [0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 4, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0]. It appears that v_3(a(n)) = 0 unless n == 0 or 2 (mod 6). (End)

Stanislav Sykora, Table of n, a(n) for n = 0..440
Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A231530 (real parts), A101686 (squares of magnitudes), A003703, A105750, A105751.

See A242651, A242652 for a pair of similar sequences.

seq(simplify(-sinh(Pi)*Im(I!*(n-I)!)/Pi), n=0..19); # Peter Luschny, Oct 23 2015

Table[Im[Pochhammer[1+I, n]], {n, 0, 20}]
Table[Sum[(-1)^(n+k) StirlingS1[n+1, 2k], {k, 0, (n+1)/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)

Factim(nmax, m)={local(a, k); a=vector(nmax); a[1]=1+0*I;
  for (k=2, nmax, a[k]=a[k-1]*(k-1+m*I); ); return(a); }
a = Factim(1000,1); imag(a)

t(n) = if( n<0, 0, n! * polcoeff(cos(log(1+x+x*O(x^n))), n));
vector(50, n, n--; (-1)^n*t(n+1)) \\ Altug Alkan, Oct 22 2015

from sympy.functions.combinatorial.numbers import stirling
def A231531(n): return sum(stirling(n+1,k<<1,kind=1)*(1 if k&1 else -1) for k in range((n+1>>1)+1)) # Chai Wah Wu, Feb 22 2024

From Vladimir Reshetnikov, Oct 22 2015: (Start)
a(n) = Im((1+i)_n) = -Re(Gamma(i)*Gamma(n+1-i))*sinh(Pi)/Pi, where (a)_n is the Pochhammer symbol, i=sqrt(-1).
a(n) = (-1)^n*A003703(n+1).
E.g.f.: sin(log(1-x))/(x-1). (End)
P-recursive: a(n) = (2*n - 1)*a(n-1) - (n^2 - 2*n + 2)*a(n-2) with a(0) = 0 and a(1) = 1. - Peter Bala, Jun 01 2023

A363416 a(n) = 1/sqrt(5) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-5).

Original entry on oeis.org

0, 1, 3, -24, -240, 1890, 40446, -311472, -12038544, 86898420, 5614173180, -36099955584, -3786960576672, 20307572439336, 3492389655843480, -14110473458954880, -4223754447793582464, 10493742733654512528, 6488421280167604253616, -4618066393756887442560, -12344309538368967592151040

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1, 3, 7 or 9 (mod 20), equivalently, rational primes that split in the field extension Q(sqrt(-5)) of Q, together with the prime p = 2. See A139513.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
(ii) for the other primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 11, 13, 17 or 19 (mod 20), equivalently, primes that remain inert in the field extension Q(sqrt(-5)) of Q, together with the prime p = 5, which ramifies in Q(sqrt(-5)). See A003626.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(a(n)) : n = 1..80] = [0, 1, 1, 1, 3, 3, 3, 4, 4, 4, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 16, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22, 22, 25, 25, 25, 26, 26, 26, 27, 27, 27, 29, 29, 29, 30, 30, 30, 31, 31, 31, 33, 33, 33, 34, 34, 34, 35, 35, 35, 39, 39, 41, 40, 40].
Note that v_3(a(80)) = 40 = 80/(3 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 11: the sequence of 11-adic valuations [v_11(a(n)) : n = 1..121] = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363415

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(5*n^2 - 10*n + 6)*a(n-2) with
a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n+1)/2)} (-5)^k*Stirling1(n+1,n-2*k).

A370549 a(n) is the numerator of the imaginary part of Product_{k=1..n} (1/k + i) where i is the imaginary unit.

Original entry on oeis.org

1, 3, 0, -5, -3, 35, 11, -65, -3055, 4433, 2561, -18863, -614635, 14705, 3871801, -6702403, -23794993, -21392575, 1677354925, 2206770805, -30432904645, -617315066615, 2099373575975, 551582580432325, -11053607615333933, -180184164588301, 4198057769186443, 435884809756010315

Offset: 1

Hugo Pfoertner, Feb 22 2024

			See A370547.

Cf. A105751, A370547, A370548, A370550.

a370549(n) = numerator(imag(prod(k=1, n, 1/k+I)))

from math import factorial, gcd
from sympy.functions.combinatorial.numbers import stirling
def A370549(n): return (a:=sum(stirling(n+1,n-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1)))//gcd(a,factorial(n)) # Chai Wah Wu, Feb 22 2024

a(n) = numerator of A105751(n)/n!. - Chai Wah Wu, Feb 22 2024

A370550 a(n) is the denominator of the imaginary part of Product_{k=1..n} (1/k + i) where i is the imaginary unit.

Original entry on oeis.org

1, 2, 1, 3, 4, 24, 9, 56, 2016, 5184, 1512, 33264, 342144, 48384, 2095632, 100590336, 12773376, 146313216, 905313024, 6552741888, 16679706624, 1216740704256, 1177309292544, 835553223622656, 6380588253118464, 226043384168448, 2506659670867968, 473758677794045952

Offset: 1

Hugo Pfoertner, Feb 22 2024

			See A370547.

Cf. A105751, A370547, A370548, A370549.

a370550(n) = denominator(imag(prod(k=1, n, 1/k+I)))

from math import factorial, gcd
from sympy.functions.combinatorial.numbers import stirling
def A370550(n): return (a:=factorial(n))//gcd(a,sum(stirling(n+1,n-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1))) # Chai Wah Wu, Feb 22 2024

a(n) = denominator of A105751(n)/n!. - Chai Wah Wu, Feb 22 2024

A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

Original entry on oeis.org

0, 1, 3, -6, -90, 45, 5607, 8316, -616572, -2517075, 106354215, 779869134, -26562900078, -299503403199, 9075456298755, 144911485323000, -4066415773786872, -87372799002303111, 2313066895842715947, 64609858869087786210, -1627745411473223627970

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 or 3 (mod 8), equivalently, rational primes that split in the field extension Q(sqrt(-2)) of Q. See A033200.
Moll's conjecture 5.5 extends to this sequence: for primes p of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 5 or 7 (mod 8), equivalently, primes that remain inert in the field extension Q(sqrt(-2)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-2)). See A033203.

			Type 2 prime p = 3: the sequence of 3-adic valuations [v_3(n) : n = 1..100] =  [0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 9, 9, 9, 10, 10, 10, 11, 11, 11, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 22, 23, 23, 23, 24, 24, 24, 26, 26, 26, 27, 27, 27, 28, 28, 28, 30, 30, 30, 31, 31, 31, 32, 32, 32, 36, 36, 36, 37, 37, 37, 38, 38, 38, 41, 40, 40, 42, 41, 41, 42, 42, 42, 44, 44, 44, 45, 45, 45, 46, 46, 46, 49, 49, 49, 50].
Note that v_3(a(100)) = 50 = 100/(3 - 1), in agrement with the asymptotic growth for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(n) : n = 1..100] = [0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else ( (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(2*n^2 - 4*n + 3)*a(n-2) with a(0) = 0 and a(1) = 1.

a(n) = Sum_{k = 0..floor((n+1)/2)} (-2)^k*Stirling1(n+1,n+1-2*k).

A363412 a(n) = 1/sqrt(3) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-3).

Original entry on oeis.org

0, 1, 3, -12, -140, 420, 13692, -23744, -2366784, 126000, 641927440, 1306329024, -252172135488, -1138135788608, 135593735484480, 999117715814400, -95707279587325952, -1013737882826462976, 85873512374909086464, 1217682899871358735360, -95486742904897158097920

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
We conjecture that in this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes of the form p == 1 (mod 6), i.e., rational primes that split in the field extension Q(sqrt(-3)) of Q, together with the prime p = 2. See A002476.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/2 as n -> oo.
(ii) for prime p == 1 (mod 6), the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes of the form p == 5 (mod 6), i.e., the rational primes that remain inert in the field extension Q(sqrt(-3)) of Q, together with the prime p = 3, which ramifies in Q(sqrt(-3)). See A007528.

			Type 2 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..60] = [0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 11, 10].
Note that v_7(a(60)) = 10 = 60/(7 - 1) in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 0..100] = [0, 0, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

Table[Im[Product[1 + k*Sqrt[-3], {k, 0, n}]] / Sqrt[3], {n, 0, 20}] (* Vaclav Kotesovec, Mar 27 2025 *)

a(n) = Sum_{k = 0..floor((n+1)/2)} (-3)^k*Stirling1(n+1,n-2*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(3*n^2 - 6*n + 4)*a(n-2) with a(0) = 0 and a(1) = 1.
Conjectures: 3 does not divide a(3*n+1) for all n; the 3-adic valuation v_3(a(3*n)) = v_3(a(3*n-1)) for all n.

A363414 a(n) = (1/2) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-4).

Original entry on oeis.org

0, 1, 3, -18, -190, 1035, 25305, -120260, -5954940, 22115925, 2197084175, -5141457750, -1173207584250, 769657081375, 856957094209125, 1127788828491000, -821262134429035000, -2922085673288364375, 1000078365473764126875, 6056214264965246443750, -1508740652939902034493750

Offset: 0

Peter Bala, Jun 01 2023

Compare with A105751(n) = the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-1).
Moll (2012) studied the prime divisors of the terms of A105750 - the real part of  Product_{k = 0..n} 1 + k*sqrt(-1) - and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
In this case, unlike in A105750, the set of type 1 primes is conjecturally empty; it appears that every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q. See A002144.
Moll's conjecture 5.5 extends to this sequence: for the primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002145.

			Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 1..100] = [0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 14, 12, 13, 12, 12, 14, 13, 14, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 19, 18, 19, 18, 18, 21, 19, 20, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..100] = [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.

Victor H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.

Cf. A105750, A105751, A363409 - A363416.

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (
(2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) )/(n - 1) end if; end:
seq(a(n), n = 0..20);

a(n) = Sum_{k = 0..floor(n/2)} (-4)^k*Stirling1(n+1,n-2*k).
P-recursive: (n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(4*n^2 - 8*n + 5)*a(n-2) with
a(0) = 0 and a(1) = 1.

cp's OEIS Frontend

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

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A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).

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A231531 Imaginary part of Product_{k = 1..n} (k + i), i = sqrt(-1).

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A363416 a(n) = 1/sqrt(5) * the imaginary part of Product_{k = 0..n} 1 + k*sqrt(-5).

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A370549 a(n) is the numerator of the imaginary part of Product_{k=1..n} (1/k + i) where i is the imaginary unit.

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A370550 a(n) is the denominator of the imaginary part of Product_{k=1..n} (1/k + i) where i is the imaginary unit.

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A363410 a(n)= 1/sqrt(2) * the imaginary part of Product_{k = 1..n} (1 + k*sqrt(-2)).

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