A101455 -id:A101455 - cp's OEIS frontend

A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

1, 256, 6560, 65536, 390626, 1679360, 5764800, 16777216, 43040161, 100000256, 214358880, 429916160, 815730722, 1475788800, 2562506560, 4294967296, 6975757442, 11018281216, 16983563040, 25600065536, 37817088000, 54875873280, 78310985280, 110058536960, 152588281251, 208827064832

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455, A258816.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, this sequence, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^8 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(8*e+8) - s[p]^(e+1))/(p^8 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321832(n)=factorback(apply(f->f[1]^(8*f[2]+8)\/(f[1]^8+f[1]%4-2),Col(factor(n)))), [1..50]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(8e+8)/(p^8 + (p mod 4) - 2)). (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = 277*Pi^9/8257536 (A258816). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^8*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.

Original entry on oeis.org

1, 512, 19682, 262144, 1953126, 10077184, 40353606, 134217728, 387400807, 1000000512, 2357947690, 5159518208, 10604499374, 20661046272, 38441425932, 68719476736, 118587876498, 198349213184, 322687697778, 512000262144, 794239673292

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, this sequence, A321834, A321835, A321836.

f:= n ->
mul(piecewise(t[1]=2,2^(9*t[2]), t[1] mod 4 = 1, (t[1]^(9*(t[2]+1))-1)/(t[1]^9-1), (t[1]^(9*(t[2]+1))+(-1)^t[2])/(t[1]^9+1)), t = ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Nov 26 2018

s[n_,r_] := DivisorSum[n, #^9 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(9*e+9) - s[p]^(e+1))/(p^9 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^9)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^9*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Robert Israel, Nov 26 2018: (Start) a(2^m) = 2^(9*m).
For prime p == 1 (mod 4), a(p^m) = (p^(9(m+1))-1)/(p^9-1).
For prime p == 3 (mod 4), a(p^m) = (p^(9(m+1))+(-1)^m)/(p^9+1). (End)
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - A101455(p)^(e+1))/(p^9 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = beta(10) = 0.99998316402... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^9*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

Original entry on oeis.org

1, 1024, 59048, 1048576, 9765626, 60465152, 282475248, 1073741824, 3486725353, 10000001024, 25937424600, 61916315648, 137858491850, 289254653952, 576640684048, 1099511627776, 2015993900450, 3570406761472, 6131066257800, 10240001048576

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

Original entry on oeis.org

1, 2048, 177146, 4194304, 48828126, 362795008, 1977326742, 8589934592, 31380882463, 100000002048, 285311670610, 743004176384, 1792160394038, 4049565167616, 8649707208396, 17592186044416, 34271896307634, 64268047284224, 116490258898218

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for related sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.

s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2.

Original entry on oeis.org

1, 1, -8, 1, 26, -8, -48, 1, 73, 26, -120, -8, 170, -48, -208, 1, 290, 73, -360, 26, 384, -120, -528, -8, 651, 170, -656, -48, 842, -208, -960, 1, 960, 290, -1248, 73, 1370, -360, -1360, 26, 1682, 384, -1848, -120, 1898, -528, -2208, -8, 2353, 651, -2320, 170

Offset: 1

N. J. A. Sloane

Multiplicative because it is the Inverse Moebius transform of [1, 0, -3^2, 0, 5^2, 0, -7^2, ...], which is multiplicative. - Christian G. Bower, May 18 2005

			The divisors of 15 are 1,3,5,15, so a(15)=(1^2+5^2)-(3^2+15^2) = -208.
G.f. = x + x^2 - 8*x^3 + x^4 + 26*x^5 - 8*x^6 - 48*x^7 + x^8 + 73*x^9 + ... - _Michael Somos_, Jun 25 2019

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J. (22) (2010), 171.
Jan Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.
Index entries for sequences mentioned by Glaisher.

Cf. A050450, A050453, A120030.

Cf. A056594.

a002173 n = a050450 n - a050453 n  -- Reinhard Zumkeller, Jun 17 2013

with(numtheory):
A002173:= proc(n)
    local count1, count3, d;
    count1 := 0:
    count3 := 0:
    for d in numtheory[divisors](n) do
        if d mod 4 = 1 then
            count1 := count1+d^2
        elif d mod 4 = 3 then
            count3 := count3+d^2
        fi:
    end do:
    count1-count3;
end proc: # Ridouane Oudra, Feb 21 2023
# second Maple program:
a:= n-> add(`if`(d::odd, d^2*(-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Ridouane Oudra, Feb 21 2023

QP = QPochhammer; s = (1-QP[q]^4*(QP[q^2]^6/QP[q^4]^4))/(4*q) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^2]^4) / 4, {q, 0, n}]; (* Michael Somos, Jun 25 2019 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^2)^(e+1)-1)/(p^2-1), ((-p^2)^(e+1)-1)/(-p^2-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */

from math import prod
from sympy import factorint
def A002173(n): return prod(((m:=p**2*(0,1,0,-1)[p&3])**(e+1)-1)//(m-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

a(n) = A050450(n) - A050453(n).
A120030(n) = -4*a(n), if n>0.
Multiplicative with a(p^e) = 1 if p = 2; ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4); ((-p^2)^(e+1)-1)/(-p^2-1) if p == 3 (mod 4). - David W. Wilson, Sep 01 2001 [This can be written as a single formula: a(p^e) = ((p^2*Chi(p))^(e+1) - 1)/(p^2*Chi(p) - 1), Chi = A101455. - Jianing Song, Oct 30 2019]
G.f.: Sum_{n>=1} A056594(n-1)*n^2*q^n/(1-q^n).
Expansion of (1 - theta_4(q)^2 * theta_4(q^2)^4)/4 in powers of q. - Michael Somos, Aug 09 2006
Expansion of (1-eta(q)^4*eta(q^2)^6/eta(q^4)^4)/4 in powers of q.
G.f.: q*G'(q)/G(q), with G(q) = Product_{n>=1} (1-q^n)^(4n*A056594(n+1)).
a(n) = Sum_{d|n} d^2*sin(d*Pi/2). - Ridouane Oudra, Feb 21 2023
G.f.: Sum_{n>=0} (4*n + 1)^2*x^(4*n + 1)/(1 - x^(4*n + 1)) - (4*n + 3)^2*x^(4*n + 3)/(1 - x^(4*n + 3)). - Miles Wilson, Oct 26 2024

More terms from David W. Wilson

A289741 a(n) = Kronecker symbol (-20/n).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Dec 27 2018

Period 20: repeat [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1].
This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and {((-100)/n)} (A185276).
Note that (Sum_{i=0..19} i*a(i))/(-20) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-5)). The fact Q(sqrt(-5)) has class number 2 implies that Q(sqrt(-5)) is not a unique factorization domain.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Cf. A035170 (inverse Moebius transform).
Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), this sequence (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (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).
Cf. A045797, A045798, A185276.

Array[KroneckerSymbol[-20, #]&, 100, 0] (* Amiram Eldar, Jan 10 2019 *)

PARI
```
a(n) = kronecker(-20, n)
```

a(n) = 1 for n in A045797; -1 for n in A045798; 0 for n that are not coprime with 20.
Completely multiplicative with a(p) = a(p mod 20) for primes p.
a(n) = A080891(n)*A101455(n).
a(n) = -a(n+10) = -a(-n) for all n in Z.
Multiplicative with a(2) = a(5) = 0, a(p) = (-1)^floor(p/10) otherwise; equivalently: a(n) = (-1)^floor(n/10) if n is coprime to 2*5, 0 otherwise. - M. F. Hasler, Feb 28 2022

A318608 Moebius function mu(n) defined for the Gaussian integers.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Aug 30 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Gaussian prime factor, otherwise (-1)^t if n is a product of a Gaussian unit and t distinct Gaussian prime factors.
a(n) = 0 for even n since 2 = -i*(1 + i)^2 contains a squared factor. For rational primes p == 1 (mod 4), p is always factored as (x + y*i)(x - y*i), x + y*i and x - y*i are not associated so a(p) = (-1)*(-1) = 1.
Interestingly, a(n) and A091069(n) have the same absolute value (= |A087003(n)|), since the discriminants of the quadratic fields Q[i] and Q[sqrt(2)] are -4 and 8 respectively, resulting in Q[i] and Q[sqrt(2)] being two of the three quadratic fields with discriminant a power of 2 or negated (the other one being Q[sqrt(-2)] with discriminant -8).

			a(15) = -1 because 15 is factored as 3*(2 + i)*(2 - i) with three distinct Gaussian prime factors.
a(21) = (-1)*(-1) = 1 because 21 = 3*7 where 3 and 7 are congruent to 3 mod 4 (thus being Gaussian primes).

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer

Absolute values are the same as those of A087003.
First row and column of A103226.
Cf. A101455.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319448.
Cf. A091069 (Moebius function over Z[sqrt(2)]).

f[p_, e_] := If[p == 2 || e > 1, 0, Switch[Mod[p, 4], 1, 1, 3, -1]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2||e>=2, r=0);
        if(Mod(p,4)==3&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - (p mod 4)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == 1 (mod 4) and -1 if p == 3 (mod 4).
a(n) = 0 if A078458(n) != A086275(n), otherwise (-1)^A086275(n).
a(n) = A103226(n,0) = A103226(0,n).
For squarefree n, a(n) = Kronecker symbol (-4, n) = A101455(n). Also for these n, a(n) = A091069(n) if n even or n == 1 (mod 8), otherwise -A091069(n).

A008442 Expansion of Jacobi theta constant (theta_2(2z))^2/4.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) is the number of ways of writing 2n as the sum of two odd positive squares. (Cf. A290081 & A008441). - Antti Karttunen, Jul 24 2017

			G.f. = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.26).

T. D. Noe, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008441, A019675.

Even bisection of A290081.

A := Basis( ModularForms( Gamma1(16), 1), 106); A[2] + 2*A[6]; /* Michael Somos, Feb 22 2015 */

a[n_] := Sum[{0, 1, -1, -1, 0, 1, 1, -1}[[Mod[d, 8] + 1]], {d, Divisors[n]}]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 15 2013, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2]^2 / 4, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] != 1, 0, Sum[ KroneckerSymbol[ 4, d], {d, Divisors @n}]]; (* Michael Somos, Feb 22 2015 *)

{a(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 24 2004 */

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -1, 0, 1, 1, -1][d%8+1]))}; /* Michael Somos, Apr 24 2004 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^8 + A)^4 / eta(x^4 + A)^2, n))}; /* Michael Somos, Apr 24 2004 */

from sympy import divisors
def A008442(n): return 0 if n&3!=1 else sum(((a:=d&3)==1)-(a==3) for d in divisors(n,generator=True)) # Chai Wah Wu, May 17 2023

Fine gives an explicit formula for a(n) in terms of the divisors of n.
a(n) = number of divisors of n of form 8n+1, 8n+5, 8n+6 minus number of divisors of form 8n+2, 8n+3, 8n+7. [I think Fine's version is simpler - N. J. A. Sloane]
G.f.: s(8)^4/(s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
Expansion of q * psi(q^4)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Feb 22 2015
Expansion of eta(q^8)^4 / eta(q^4)^2 in powers of q.
Euler transform of period 8 sequence [ 0, 0, 0, 2, 0, 0, 0, -2, ...]. - Michael Somos, Apr 24 2004
a(n)=0 unless n=4k+1 in which case a(n) is the difference between number of divisors of n of form 4k+1 and 4k+3.
Multiplicative with a(2^e) = 0^e, a(p^e) = (1 + (-1)^e)/2 if p==3 mod 4 otherwise a(p^e) = 1+e. - Michael Somos, Sep 18 2004
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Sep 02 2005
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k - 1) / (1 + x^(4*k - 2)). - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 - x^(8*k)) = x Product_{k>0} (1 - x^(8*k))^4 / (1 - x^(4*k))^2. - Michael Somos, Apr 24 2004
a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/8 = 0.392699... (A019675). - Amiram Eldar, Oct 23 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s) = L(chi_s)*beta(s), where chi_1 = A000035 and chi_{-1} = A101455 are respectively the principal and the non-principal Dirichlet character modulo 4, and beta(s) is the Dirichlet beta function. For the formula of the sequence whose Dirichlet g.f. is Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k, see A378006. This sequence is the case k = 4. - Jianing Song, Nov 13 2024

A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1

Offset: 1

Emeric Deutsch, Nov 24 2003

Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009
Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017
From Roger Ford, Oct 17 2018: (Start)
T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.
Examples:    /\         semi-meander with 5 top arches
            //\\   /\   2 arches are at depth=0 (no covering arches)
           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)
      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)
       2, 2, 1 is the listing for this t(5,2)
             /\      semi-meander with 5 top arches
            /  \     (0)(1)
     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)
a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,   1;
  1,  2,  4,   2,   1;
  1,  3,  6,   6,   3,    1;
  1,  3,  9,   9,   9,    3,    1;
  1,  4, 12,  18,  18,   12,    4,    1;
  1,  4, 16,  24,  36,   24,   16,    4,    1;
  1,  5, 20,  40,  60,   60,   40,   20,    5,    1;
  1,  5, 25,  50, 100,  100,  100,   50,   25,    5,    1;
  1,  6, 30,  75, 150,  200,  200,  150,   75,   30,    6,   1;
  1,  6, 36,  90, 225,  300,  400,  300,  225,   90,   36,   6,   1;
  1,  7, 42, 126, 315,  525,  700,  700,  525,  315,  126,  42,   7,  1;
  1,  7, 49, 147, 441,  735, 1225, 1225, 1225,  735,  441, 147,  49,  7, 1;
  1,  8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
  ...
a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
U=(1,1), D=(1,-1).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.

[(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
Table[T[n, k], {n,13}, {k,n}]//Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022

T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)

Keyword:tabl added Philippe Deléham, Jan 25 2010

A322796 a(n) = Kronecker symbol (6/n).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Dec 26 2018

Period 24: repeat [0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1].
Also a(n) = Kronecker symbol (24/n).
This sequence is one of the seven non-principal real Dirichlet characters modulo 24. The other six are Jacobi or Kronecker symbols {(-6/n)} (or {(n/6)}, {(-24/n)}, {(n/24)}, A109017), {(-12/n)} (or {(n/12)}, A134667), {(12/n)} (A110161), {(-18/n)} (or {(-72/n)}), {(18/n)} (or {(72/n)}, {(n/72)}) and {(-36/n)}. These sequences all become the same after taking absolute values.

Antti Karttunen, Table of n, a(n) for n = 0..65543
Eric Weisstein's World of Mathematics, Kronecker Symbol (contains this sequence)
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-1).

Cf. A035188 (inverse Moebius transform).

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), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), this sequence (d=24).

[KroneckerSymbol(6, n): n in [0..100]]; // Vincenzo Librandi, Jan 01 2019

Array[KroneckerSymbol[6, #] &, 105, 0] (* Michael De Vlieger, Dec 31 2018 *)
Table[KroneckerSymbol[6, n], {n, 0, 100}] (* Vincenzo Librandi, Jan 01 2019 *)

a(n) = kronecker(6, n); \\ --- Argument order corrected by Antti Karttunen, Sep 27 2019

a(n) = 1 for n == 1, 5, 19, 23 (mod 24); -1 for n == 7, 11, 13, 17 (mod 24); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 24) for primes p.
a(n) = A091337(n)*A102283(n).
a(n) = A109017(n+12) = A109017(n-12).
a(n) = a(-n) = a(n+24) for all n in Z.

Definition corrected by Antti Karttunen, Sep 28 2019

cp's OEIS Frontend

A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

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A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.

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A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

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A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

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A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2.

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A289741 a(n) = Kronecker symbol (-20/n).

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A318608 Moebius function mu(n) defined for the Gaussian integers.

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