A009454 -id:A009454 - cp's OEIS frontend

A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 0

Gerald McGarvey, Jan 20 2005

Called X(n) (i.e., Chi(n)) in Hardy and Wright (p. 241), who show that X(n*m) = X(n)*X(m) for all n and m (i.e., X(n) is completely multiplicative) since (n*m - 1)/2 - (n - 1)/2 - (m - 1)/2 = (n - 1)*(m - 1)/2 == 0 (mod 2) when n and m are odd.
Same as A056594 but with offset 1.
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the non-principal Dirichlet character mod 4. (The principal character is A000035.)
Associated Dirichlet L-functions are for example L(1,chi) = Sum_{n>=1} a(n)/n = A003881, or L(2,chi) = Sum_{n>=1} a(n)/n^2 = A006752, or L(3,chi) = Sum_{n>=1} a(n)/n^3 = A153071. (End)
a(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 0, y = -1, z is arbitrary. - Michael Somos, Nov 27 2019

			G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=4, Chi_2(n).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241.

Muniru A Asiru, Table of n, a(n) for n = 0..1000 (a(0) prepended by Jianing Song)
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A002654, A009116, A009454, A056594, A108520, A111335, A126120, A146559.

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), this sequence (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).

a := [1, 0];; for n in [3..10^2] do a[n] := a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // G. C. Greubel, Aug 23 2018

a := n -> `if`(n mod 2=0, 0, (-1)^((n-1)/2)):
seq(a(n), n=1..10^3); # Muniru A Asiru, Feb 02 2018

a[ n_] := {1, 0, -1, 0}[[ Mod[ n, 4, 1]]]; (* Michael Somos, Jan 13 2014 *)
LinearRecurrence[{0, -1}, {1, 0}, 75] (* G. C. Greubel, Aug 23 2018 *)

{a(n) = if( n%2, (-1)^(n\2))}; /* Michael Somos, Sep 02 2005 */

{a(n) = kronecker( -4, n)}; /* Michael Somos, Mar 30 2012 */

def A101455(n): return (0,1,0,-1)[n&3] # Chai Wah Wu, Jun 21 2024

Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - Mitch Harris May 17 2005
Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos, Sep 02 2005
G.f.: (x - x^3)/(1 - x^4) = x/(1 + x^2). - Michael Somos, Sep 02 2005
G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 * (1 + 2*v). - Michael Somos, Aug 04 2011
a(n + 4) = a(n), a(n + 2) = a(-n) = -a(n), a(2*n) = 0, a(2*n + 1) = (-1)^n for all n in Z. - Michael Somos, Aug 04 2011
a(n + 1) = A056594(n). - Michael Somos, Jan 13 2014
REVERT transform is A126120. STIRLING transform of A009454. BINOMIAL transform is A146559. BINOMIAL transform of A009116. BIN1 transform is A108520. MOBIUS transform of A002654. EULER transform is A111335. - Michael Somos, Mar 30 2012
Completely multiplicative with a(p) = 2 - (p mod 4). - Werner Schulte, Feb 01 2018
a(n) = (-(n mod 2))^binomial(n, 2). - Peter Luschny, Sep 08 2018
a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - Jianing Song, Sep 09 2018
From Jianing Song, Nov 14 2018: (Start)
a(n) = ((-4)/n) (or more generally, ((-4^i)/n) for i > 0), where (k/n) is the Kronecker symbol.
E.g.f.: sin(x).
Dirichlet g.f. is the Dirichlet beta function.
a(n) = A091337(n)*A188510(n). (End)

a(0) prepended by Jianing Song, Nov 14 2024

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

Original entry on oeis.org

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

A003703 Expansion of e.g.f. cos(log(1+x)).

Original entry on oeis.org

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

Offset: 0

R. H. Hardin, Simon Plouffe

			1 - x^2 + 3*x^3 - 10*x^4 + 40*x^5 - 190*x^6 + 1050*x^7 - 6620*x^8 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 100 terms from T. D. Noe)
Vaclav Kotesovec, Graph a(n+1)/a(n)
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A009024, A009454.

a:= n-> add(Stirling1(n, 2*k) * (-1)^(k), k=0..floor(n/2)):
seq(a(n), n=0..20);

CoefficientList[Series[Cos[Log[1 + x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 16 2015 *)
Table[(-1)^n Im[Pochhammer[1-I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)

{a(n) = if( n<0, 0, n! * polcoeff( cos( log( 1 + x + x * O(x^n))), n))} /* Michael Somos, Jul 26 2012 */

a(n) = (-1)^n*(prod(k=0, n-1, I+k)+prod(k=0, n-1, -I+k))/2; \\ Seiichi Manyama, Oct 10 2022

from sympy.functions.combinatorial.numbers import stirling
def A003703(n): return sum(stirling(n,k<<1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = Sum_{k=0..n-1} (-1)^(k+1)*T(n-k, k)*sin(Pi*(n-k-1)/2) + 0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry, Apr 18 2005
abs(a(n)) = abs(f(n)) with f(n)=Product_{k=1..n} i+k (where i^2=-1). - Yalcin Aktar, Jul 13 2009
a(n) = Sum_{k=0..floor(n/2)} Stirling1(n,2*k)*(-1)^k. - Vladimir Kruchinin, Jan 29 2011
a(n+2)= -a(n+1)*(2*n+1) - a(n)*(1+n^2), a(0)=1, a(1)=0. - Sergei N. Gladkovskii, Aug 17 2012
a(n) = (-1)^n * ( (i)n + (-i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit. - _Seiichi Manyama, Oct 10 2022
a(n) = Re(gamma(i+1)/gamma(i+1-n)). The imaginary part is A009454. - Colin Beveridge, Jul 30 2024

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

A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

Original entry on oeis.org

0, 1, 3, 0, -40, -90, 1050, 6160, -46800, -549900, 3103100, 67610400, -271627200, -11186357000, 26495469000, 2416003824000, -1394099824000, -662595375078000, -936096296850000, 225382826562400000, 819329864480400000, -93217812901913700000, -570263312237604700000

Offset: 0

Paul Barry, Apr 18 2005

From Peter Bala, Jun 01 2023: (Start)
Compare with A105750(n) = the real part of Product_{k = 0..n} (1 + k*sqrt(-1)). Moll (2012) studied the prime divisors of the terms of A105750 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 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 (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.
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 == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q. See A002145. (End)

			From _Peter Bala_, Jun 01 2023: (Start)
The sequence of 5-adic valuations [v_5(a(n)) : n = 4..100] = [1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 12, 11, 11, 13, 11, 12, 13, 13, 12, 12, 14, 13, 13, 14, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18, 20, 19, 19, 20, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 25, 25, 24, 24, 25, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior conjectured above.
The sequence of 3-adic valuations [v_3(a(n)) : n >= 4] begins [0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, 1, 0, 3, 3, 0, 1, 3, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, ...], exhibiting the oscillatory behavior for type 3 primes conjectured above. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A003703, A009454, A048994, A105750, A164652, A231531, A363409 - A363416.

a:= proc(n) option remember; `if`(n<2, n,
      ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 11 2018

Table[Im[Product[1+k*I,{k,0,n}]],{n,0,22}] (* James C. McMahon, Jan 27 2024 *)

a(n) = imag(prod(k=0, n, 1+k*I)); \\ Michel Marcus, Apr 11 2018

from sympy.functions.combinatorial.numbers import stirling
def A105751(n): return 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) = ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1) for n > 1, a(n) = n for n < 2. - Alois P. Heinz, Apr 11 2018
From Peter Bala, May 27 2023:(Start)
a(n) = Sum_{k = 0..floor((n+1)/2)} (-1)^k*|Stirling1(n+1, n-2*k)|, where Stirling1(n, k) = A048994(n,k).
The triangular number n*(n+1)/2 divides a(n). See A164652. In particular, if p is an odd prime then p divides a(p).
a(2*n) = (-1)^(n+1)*A003703(2*n+1) for n >= 0.
a(2*n+1) = (-1)^(n+1)*A009454(2*n+2) for n >= 0. (End)

A231530 Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.

Original entry on oeis.org

1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, -67610400, -839594600, -11186357000, -159300557000, -2416003824000, -38894192662000, -662595375078000, -11911522255750000, -225382826562400000, -4477959179352100000, -93217812901913700000, -2029107997508660900000

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), where i is the imaginary unit. Hence n! = factim(n,0), while the current sequence lists the real parts of factim(n,1). The imaginary parts are in A231531 and squares of magnitudes are in A101686.

			factim(5,1) = -90 + 190*i. Hence a(5) = -90.

Stanislav Sykora, Table of n, a(n) for n = 0..440

Cf. A231531 (imaginary parts), A101686 (squares of magnitudes), A009454.

See A242651, A242652 for a pair of similar sequences.

seq(simplify(Re(I!*(n-I)!)*sinh(Pi)/Pi),n=0..22); # Peter Luschny, Oct 23 2015

Table[Re[Product[k+I,{k,n}]],{n,0,30}] (* Harvey P. Dale, Aug 04 2016 *)

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); real(a)

from sympy.functions.combinatorial.numbers import stirling
def A231530(n): return sum(stirling(n+1,(k<<1)+1,kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

A231530 = lambda n : rising_factorial(1-I, n).real()
[A231530(n) for n in range(24)] # Peter Luschny, Oct 23 2015

From Peter Luschny, Oct 23 2015: (Start)
a(n) = Re(i!*(n-i)!)*sinh(Pi)/Pi.
a(n) = n!*[x^n](cos(log(1-x))/(1-x)).
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n+k)*Stirling1(n+1,2*k+1).
a(n) = Re(rf(1+i,n)) where rf(k,n) is the rising factorial and i the imaginary unit.
a(n) = (-1)^n*A009454(n+1). (End)

A051893 a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.

Original entry on oeis.org

1, 1, 5, 50, 850, 22100, 817700, 40885000, 2657525000, 217917050000, 22009622050000, 2685173890100000, 389350214064500000, 66189536390965000000, 13039338669020105000000, 2946890539198543730000000, 757350868574025738610000000, 219631751886467464196900000000

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

Cf. A001710, A101686, A256019, A256020.

a := n -> `if`(n=1,1,(sinh(Pi)*GAMMA(n-I)*GAMMA(n+I))/(2*Pi)):
seq(simplify(a(n)), n=1..18); # Peter Luschny, Oct 19 2016

a[n_] := Pochhammer[2-I, n-2]*Pochhammer[2+I, n-2]; a[1] = 1; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 21 2012, after Vladeta Jovovic *)
Join[{1},FoldList[Times,1,Range[2,20]^2+1]] (* Harvey P. Dale, Jul 04 2013 *)
Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[i^2*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}] (* Vaclav Kotesovec, Mar 13 2015 *)

a(n) = Product_{i=2..n-1} (i^2+1), for n>2. - Vladeta Jovovic, Nov 26 2002
From Vaclav Kotesovec, Mar 13 2015: (Start)
For n > 1, a(n) = A101686(n-1)/2.
a(n) ~ (n-1)!^2 * sinh(Pi)/(2*Pi).
(End)
a(n) = (A003703(n)^2 + A009454(n)^2 + A000007(n-1))/2. - Vladimir Reshetnikov, Oct 15 2016
a(n) = sinh(Pi)*Gamma(n-I)*Gamma(n+I)/(2*Pi) for n>1. - Peter Luschny, Oct 19 2016

More terms from Harvey P. Dale, Jul 04 2013

A242652 Imaginary part of Product_{k=0..n} (i-k), where i=sqrt(-1).

Original entry on oeis.org

1, -1, 1, 0, -10, 90, -730, 6160, -55900, 549900, -5864300, 67610400, -839594600, 11186357000, -159300557000, 2416003824000, -38894192662000, 662595375078000, -11911522255750000, 225382826562400000, -4477959179352100000, 93217812901913700000, -2029107997508660900000, 46099220630461596000000

Offset: 0

N. J. A. Sloane, May 29 2014

sign

Shifted version of A009454. - R. J. Mathar, May 30 2014

			Table of n, Product_{k=0..n} (i-k):
   0, i
   1, -1 - i
   2, 3 + i
   3, -10
   4, 40 - 10*i
   5, -190 + 90*i
   6, 1050 - 730*i
   7, -6620 + 6160*i
   8, 46800 - 55900*i
   9, -365300 + 549900*i
  10, 3103100 - 5864300*i
  11, -28269800 + 67610400*i
  12, 271627200 - 839594600*i

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 561-562, Section 148.

Cf. A009454.
Cf. A242651, A101686.
A231531 is the same except for signs.

a:= n-> Im(mul(I-j, j=0..n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 03 2021

a(n) = imag(prod(k=0, n, I-k)); \\ Michel Marcus, Jan 03 2021

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A189815, A296980, A296981, A296982.

a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

a(n) ~ -(-1)^n * n^(n-1) / (exp(1) - 1)^(n - 1/2). - Vaclav Kotesovec, Mar 26 2019

A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A296435, A296979, A296981, A296982.

a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

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

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A003703 Expansion of e.g.f. cos(log(1+x)).

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

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A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

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A231530 Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.

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