A008441 -id:A008441 - cp's OEIS frontend

A005883 Theta series of square lattice with respect to deep hole.

4, 8, 4, 8, 8, 0, 12, 8, 0, 8, 8, 8, 4, 8, 0, 8, 16, 0, 8, 0, 4, 16, 8, 0, 8, 8, 0, 8, 8, 8, 4, 16, 0, 0, 8, 0, 16, 8, 8, 8, 0, 0, 12, 8, 0, 8, 16, 0, 8, 8, 0, 16, 0, 0, 0, 16, 12, 8, 8, 0, 8, 8, 0, 0, 8, 8, 16, 8, 0, 8, 8, 0, 12, 8, 0, 0, 16, 0, 8, 8, 0, 24, 0, 8, 8, 0, 0, 8, 8, 0, 4, 16, 8, 8, 16, 0, 0

Offset: 0

N. J. A. Sloane

nonn

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

In [Jacobi 1829] on page 105 is equation 18: "2 k K / Pi = 4 sqrt(q) + 8 sqrt(q^5) + 4 sqrt(q^9) [...]". - Michael Somos, Sep 09 2012

			4 + 8*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^6 + 8*x^7 + 8*x^9 + 8*x^10 + 8*x^11 + ...
4*q + 8*q^3 + 4*q^5 + 8*q^7 + 8*q^9 + 12*q^13 + 8*q^15 + 8*q^19 + 8*q^21 + ...
Theta = 4*q^(1/2) + 8*q^(5/2) + 4*q^(9/2) + 8*q^(13/2) + 8*q^(17/2) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, 1829.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441.

a[0] = 4; a[n_] := 4*DivisorSum[4n+1, (-1)^Quotient[#, 2]&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 04 2015, translated from PARI *)
s = 4*(QPochhammer[q^2]^4/QPochhammer[q]^2)+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

{a(n) = if( n<0, 0, n = 4*n + 1; 4 * sumdiv(n, d, (-1)^(d\2)))} /* Michael Somos, Oct 31 2006 */

Expansion of Jacobi theta constant q^(-1/2) * theta_2(q)^2 in powers of q^2. - Michael Somos, Oct 31 2006
G.f.: 4 * (Product_{k>0} (1 - x^k) * (1 + x^(2*k))^2)^2. - Michael Somos, Oct 31 2006
From Michael Somos, Sep 09 2012: (Start)
Expansion of 4 * psi(x)^2 in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(-1) * (1/2) * (1 - k') * K / (Pi/2) in powers of q^4 where k', K are Jacobi elliptic functions.
Expansion of q^(-1/2) * k * K / (Pi/2) in powers of q^2 where k, K are Jacobi elliptic functions.
Expansion of q^(-1/4) * 2 * k^(1/2) * K / (Pi/2) in powers of q where k, K are Jacobi elliptic functions.
Expansion of 4 * q^(-1/4) * eta(q^2)^4 / eta(q)^2 in powers of q.
a(n) = 4 * A008441(n). (End)

A113406 Half the number of integer solutions to x^2 + 4 * y^2 = n.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 28 2005

			x + 2*x^4 + 2*x^5 + 2*x^8 + x^9 + 2*x^13 + 2*x^16 + 2*x^17 + 4*x^20 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003881, A004018, A004531, A008441.

s = (EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^4] - 1)/(2 q) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, qfrep( [1, 0; 0, 4], n)[n])}

{a(n) = if( n<1, 0, if( n%4==1, sumdiv( n, d, (-1)^(d\2)), if( n%4==0, 2 * sumdiv( n, d, kronecker( -4, d)))))}

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==2, 2 * (e>1), if( p%4==3, (1 + (-1)^e) / 2, e+1)))))}

a(n) is multiplicative with a(2) = 0, a(2^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4)
G.f.: (theta_3(q) * theta_3(q^4) - 1) / 2.
a(4*n + 2) = a(4*n + 3) = 0. A004531(n) = 2 * a(n) if n>0. a(4*n + 1) =  A008441(n). A004018(n) = 2 * a(4*n) if n>0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 = 0.785398... (A003881). - Amiram Eldar, Oct 15 2022

A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 0, 2, 2, 4, 0, 2, 4, 0, 4, 2, 4, 2, 0, 4, 0, 0, 0, 2, 6, 4, 2, 0, 4, 4, 0, 2, 0, 4, 0, 2, 4, 0, 4, 4, 4, 0, 0, 0, 4, 0, 0, 2, 2, 6, 4, 4, 4, 2, 0, 0, 0, 4, 0, 4, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 2, 4, 4, 6, 0, 0, 4, 0, 4, 2, 4, 0, 0, 8, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 2, 4, 2, 0, 6, 4, 4, 0, 4

Offset: 0

Michael Somos, Sep 14 2006

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

			G.f. = 1 + 2*q + 2*q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 2*q^6 + 2*q^8 + 2*q^9 + 4*q^10 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 197, Entry 44.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A008441, A019693, A035154, A113446, A125079, A125061, A132003.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Jul 09 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%12<6, e+1, !(e%2) )))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

Expansion of eta(q^2)^3 * eta(q^3)^2 * eta(q^6) / (eta(q)^2 * eta(q^4)* eta(q^12)) in powers of q.
Expansion of 2 * psi(q) * psi(q^2) * psi(q^3) / psi(q^6) - phi(q^3)^2 in powers of q. - Michael Somos, Jul 09 2013
Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].
Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].
a(12*n + 7) = a(12*n + 11) = 0.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125061.
A035154(n) = a(n) / 2 if n > 0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.
a(n) = (-1)^n * A132003(n). Expansion of (phi(-q^3) / phi(-q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.0943951... (A019693). - Amiram Eldar, Nov 21 2023

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

Original entry on oeis.org

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).

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -14, 81, -238, 322, 0, -429, 82, 0, 2162, -3038, -1134, 2401, 2482, 0, -6958, 3332, 0, 1442, 0, 6561, -4508, -9758, 0, -1918, 18802, 0, 9362, -24638, -19278, 14641, 14756, 0, 0, 6562, 0, -1148, -33998, 26082, -20398, 0, 0, 28083, 49042, 0, -64078, -30268

Offset: 0

Michael Somos, Aug 12 2012

sign

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

This is a member of an infinite family of integer weight level 8 modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

			1 - 14*x + 81*x^2 - 238*x^3 + 322*x^4 - 429*x^6 + 82*x^7 + 2162*x^9 + ...
q - 14*q^5 + 81*q^9 - 238*q^13 + 322*q^17 - 429*q^25 + 82*q^29 + 2162*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A030212, A209942, A215601.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^14 / QPochhammer[ x^2]^4, {x, 0, n}] (* Michael Somos, Sep 05 2013 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^7 / eta(x^2 + A)^2 )^2, n))}

Expansion of q^(-1/4) * eta(q)^14 / eta(q^2)^4 in powers of q.
Expansion of q^(-1/4) * ( eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 + 4 * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^4 ) in powers of q. - Michael Somos, Sep 05 2013
Euler transform of period 2 sequence [ -14, -10, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 128 (t/i)^5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030212.
a(n) = (-1)^n * A209942(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n).
a(n) = A030212(4*n + 1). - Michael Somos, Sep 05 2013

A215601 Expansion of phi(-x)^2 * f(-x)^6 + 32 * x * psi(-x)^2 * f(-x^4)^6 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 22, -27, -18, -94, 0, 359, -130, 0, 214, -230, -594, -343, 518, 0, 830, -396, 0, 1098, 0, 729, -2068, -1670, 0, 594, 598, 0, -1746, 2002, 486, -1331, 5148, 0, 0, -1606, 0, -2860, -3514, 2538, 286, 0, 0, -1873, -4082, 0, 3942, 4708, 0, 5362, 1174, 0, -5060

Offset: 0

Michael Somos, Aug 16 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by g_4(q) in Cynk and Hulek on page 8 as the unique weight 4 Hecke eigenform of level 32 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

			G.f. = 1 + 22*x - 27*x^2 - 18*x^3 - 94*x^4 + 359*x^6 - 130*x^7 + 214*x^9 - 230*x^10 + ..
G.f. = q + 22*q^5 - 27*q^9 - 18*q^13 - 94*q^17 + 359*q^25 - 130*q^29 + 214*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A215472, A215600.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2 + 32 x (QPochhammer[ x] QPochhammer[ x^4]^4 / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jan 11 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^5 / eta(x^2 + A) )^2 + 32 * x * ( eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A) )^2, n))};

{a(n) = local(A, p, e, x, y, a0, a1, w=3); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4==3, if( e%2, 0, (-p)^(w*e/2)), y=-sum( i=0, p-1, kronecker( i^3-i, p)); a0=2; a1=y; for( i=2, w, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for( i=2, e, x=y*a1 -p^w*a0; a0=a1; a1=x); a1)))))};

Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 + 32 * (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^10 (t/i)^4 f(t) where q = exp(2 Pi i t).
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -27 * a(n). a(n) = A215600(2*n).

A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 31 2024

nonn

Cf. A008441, A033687, A097195, A340456.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+1))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-2))*(1-x^(7*k-5))/((1-x^(7*k-1))*(1-x^(7*k-6)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-2)) * (1-x^(7*k-5)) / ((1-x^(7*k-1)) * (1-x^(7*k-6)))^2.

A129447 Expansion of psi(q) * psi(q^3) * phi(q^3) / phi(q) in powers of q where psi(), phi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 16 2007

sign

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

			G.f. = 1 - x + 2*x^2 + x^4 + 2*x^6 - 2*x^7 + 2*x^8 + 3*x^12 - x^13 + 2*x^14 + ...
G.f. = q - q^3 + 2*q^5 + q^9 + 2*q^13 - 2*q^15 + 2*q^17 + 3*q^25 - q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A008441, A121444, A125079.

a[ n_] := If[ n < 0, 0, Module[ {m = n}, If[ Mod[n, 6] == 1, m = Quotient[ n, 3]; -1, 1] DivisorSum[ 2 m + 1, KroneckerSymbol[ -4, #] &]]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Which[# == 1, 1, # == 2, 0, # == 3, (-1)^#2, Mod[#, 4] == 1, #2 + 1, True, Mod[#2 + 1, 2]] & @@@ FactorInteger[2 n + 1])]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<0, 0, if( n%6==1, n\=3; -1, 1) * sumdiv(2*n + 1, d, kronecker(-4, d)) )};

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, (-1)^e, p%4==1, e+1, 1-e%2 )))};

Expansion of q^(-1/2) * eta(q) * eta(q^4)^2 * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 0, -1, -2, -1, 0, 2, 2, -1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f.: Product_{k>0} (1 + x^(2*k))^2 * (1 - x^(3*k))^2 * (1 + x^(3*k))^5 / ((1 + x^k) * (1 + x^(6*k))^2).
G.f.: Sum_{k in Z} x^(3*k) / (1 + x^(6*k + 1)) = Sum_{k>0} x^(k-1) * (1 - x^(2*k -1))^2 / (1 + x^(6*k - 3)).
abs(a(n)) = A125079(n). a(6*n + 3) = a(6*n + 5) = 0.
a(6*n) = A002175(n). a(2*n) = A008441(n). a(6*n + 1) = - A008441(n). a(6*n + 2) = 2* A121444(n).

A035181 a(n) = Sum_{d|n} Kronecker(-9, d).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 0, 4, 1, 4, 0, 3, 2, 0, 2, 5, 2, 2, 0, 6, 0, 0, 0, 4, 3, 4, 1, 0, 2, 4, 0, 6, 0, 4, 0, 3, 2, 0, 2, 8, 2, 0, 0, 0, 2, 0, 0, 5, 1, 6, 2, 6, 2, 2, 0, 0, 0, 4, 0, 6, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 3, 0, 0, 4, 0, 10, 1, 4, 0, 0, 4, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 9, 2, 4, 0, 8, 0

Offset: 1

N. J. A. Sloane

			x + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 4*x^8 + x^9 + 4*x^10 + 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A008441, A019693, A125079.

Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), A035184 (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -9, d], { d, Divisors[ n]}]] (* Michael Somos, Jun 24 2011 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -9, d)))} \\ Michael Somos, Jun 24 2011

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -9, p) * X))) [n])} \\ Michael Somos, Jun 24 2011

{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==2, e+1, if( p==3, 1, if( p%4==1, e+1, (1 + (-1)^e)/2))))))} \\ Michael Somos, Jun 24 2011

A035181(n)=sumdivmult(n,d,kronecker(-9,d)) \\ M. F. Hasler, May 08 2018

From Michael Somos, Jun 24 2011: (Start)
a(n) is multiplicative with a(2^e) = e + 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4) and p > 3.
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-9, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-9, p) * p^-s)). (End)
a(3*n) = a(n). a(2*n + 1) = A125079(n). a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.094395... (A019693). - Amiram Eldar, Oct 17 2022

cp's OEIS Frontend

A005883 Theta series of square lattice with respect to deep hole.

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A113406 Half the number of integer solutions to x^2 + 4 * y^2 = n.

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A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

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

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A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

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A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A215601 Expansion of phi(-x)^2 * f(-x)^6 + 32 * x * psi(-x)^2 * f(-x^4)^6 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

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