A378006 -id:A378006 - cp's OEIS frontend

A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 0, 2, 0, 0, 2, 1, 1, 1, 0, 4, 0, 1, 0, 3, 0, 1, 1, 1, 4, 6, 2, 6, 2, 4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 1, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1

Offset: 0

Jianing Song, Nov 14 2024

A condensed version of A378006: the k-th column is the sequence {b(k*n+1)}, with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

			Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 1, 2, 0, 2, 2, 2, 0, 4, ...
  1, 1, 2, 1, 4, 2, 0, 4, 6, 0, ...
  1, 1, 0, 2, 1, 2, 0, 2, 0, 4, ...
  1, 1, 2, 2, 0, 1, 6, 0, 6, 4, ...
  1, 1, 1, 0, 0, 2, 0, 4, 0, 0, ...
  1, 1, 2, 3, 4, 2, 6, 2, 0, 4, ...
  1, 1, 0, 2, 0, 2, 0, 0, 1, 4, ...
  1, 1, 1, 0, 4, 3, 0, 0, 6, 1, ...
  1, 1, 2, 2, 0, 0, 3, 4, 0, 0, ...
  1, 1, 2, 2, 0, 2, 6, 3, 0, 4, ...
Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2.
Column k = 1: 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s + 1/8^s + 1/9^s + 1/10^s + 1/11^s + ...;
Column k = 2: 1 + 1/3^s + 1/5^s + 1/7^s + 1/9^s + 1/11^s + 1/13^s + 1/15^s + 1/17^s + 1/19^s + 1/21^s + ...;
Column k = 3: (1 + 1/2^s + 1/4^s + 1/5^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + ...) = 1 + 1/4^s + 2/7^s + 2/13^s + 1/16^s + 2/19^s + 1/25^s + 2/28^s + 2/31^s + ...;
Column k = 4: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...) = 1 + 2/5^s + 1/9^s + 2/13^s + 2/17^s + 3/25^s + 2/29^s + 2/37^s + 2/41^s + ...;
Column k = 5: (1 + 1/2^s + 1/3^s + 1/4^s + ...)*(1 + i/2^s - i/3^s - 1/4^s + ...)*(1 - 1/2^s - 1/3^s + 1/4^s + ...)*(1 - i/2^s + i/3^s - 1/4^s + ...) = 1 + 4/11^s + 1/16^s + 4/31^s + 4/41^s + ...;
Column k = 6: (1 + 1/5^s + 1/7^s + 1/11^s + ...)*(1 - 1/5^s + 1/7^s - 1/11^s + ...) = 1 + 2/7^s + 2/13^s + 2/19^s + 1/25^s + 1/31^s + 2/37^s + 2/43^s + 3/49^s + 2/61^s + ...;
Column k = 7: (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...;
Column k = 8: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 + 1/3^s - 1/5^s - 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...)*(1 - 1/3^s - 1/5^s + 1/7^s + ...) = 1 + 2/9^s + 4/17^s + 2/25^s + 4/41^s + 2/49^s + 4/73^s + 3/81^s + ...;
Column k = 9: (1 + 1/2^s + 1/4^s + 1/5^s + 1/7^s + 1/8^s + ...)*(1 + (w+1)/2^s + w/4^s - w/5^s - (w+1)/7^s - 1/8^s + ...)*(1 + w/2^s - (w+1)/4^s - (w+1)/5^s + w/7^s + 1/8^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + 1/7^s - 1/8^s + ...)*(1 - (w+1)/2^s + w/4^s + w/5^s - (w+1)/7^s + 1/8^s + ...)*(1 - w/2^s - (w+1)/4^s + (w+1)/5^s + w/7^s - 1/8^s + ...) = 1 + 6/19^s + 6/37^s + 1/64^s + 6/73^s + ...;
Column k = 10: (1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...

Jianing Song, Table of n, a(n) for n = 0..11324 (the first 150 diagonals, with n+k = 1..150)

Columns: A000012 (k=1 and k=2), A033687 (k=3), A008441 (k=4), A378008 (k=5), A097195 (k=6), A378009 (k=7), A378010 (k=8), A378011 (k=9), A378012 (k=10).

Cf. A378006.

A378007(n,k) = {
my(f = factor(k*n+1), res = 1); for(i=1, #f~, my(d = znorder(Mod(f[i,1],k)));
if(f[i,2] % d != 0, return(0), my(m = f[i,2]/d, r = eulerphi(k)/d); res *= binomial(m+r-1,r-1)));
res;}

See A378006.

For odd k, T(2*k,n) = T(k,2*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

A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + ...

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A033685, A033687, A097195, A217219.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^9]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Feb 22 2015 *)
f[p_, e_] := If[Mod[p, 3] == 1, e + 1, (1 + (-1)^e)/2]; f[3, e_] := 0; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Oct 13 2022 *)

{a(n) = local(A, p, e); if( n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p!=3, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 25 2005

{a(n) = local(A); if( (n<1) || (n%3!=1), 0, n = (n-1)/3; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, May 25 2005

From Michael Somos, May 25 2005: (Start)
Euler transform of period 9 sequence [ 0, 0, 1, 0, 0, 1, 0, 0, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3.
a(3*n) = a(3*n + 2) = 0. a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). 3*a(n) = A033685(n).
Multiplicative with a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(9*sqrt(3)) = 0.403066... . - Amiram Eldar, Oct 13 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s), where chi_1 = A011655 and chi_{-1} = A102283 are respectively the principal and the non-principal Dirichlet character modulo 3. 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 = 3. - Jianing Song, Nov 13 2024

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A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

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

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A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.

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