A002652 -id:A002652 - cp's OEIS frontend

A138811 Theta series of quadratic form x^2 + xy + 11y^2.

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

Offset: 0

Michael Somos, Mar 31 2008, Apr 05 2008

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

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035147.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), this sequence (d=-43).

A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* Michael Somos, Sep 07 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
2 Table[a[n], {n, 1, 100}]] (* Vincenzo Librandi, Sep 07 2018 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};

a(n)=if(n, sumdivmult(n,d,kronecker(-43,d))*2, 0) \\ Charles R Greathouse IV, Nov 23 2021

Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
a(n) = 2 * A035147(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - Amiram Eldar, Nov 21 2023

A028594 Expansion of (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2 in powers of q.

Original entry on oeis.org

1, 4, 12, 16, 28, 24, 48, 4, 60, 52, 72, 48, 112, 56, 12, 96, 124, 72, 156, 80, 168, 16, 144, 96, 240, 124, 168, 160, 28, 120, 288, 128, 252, 192, 216, 24, 364, 152, 240, 224, 360, 168, 48, 176, 336, 312, 288, 192

Offset: 0

N. J. A. Sloane

nonn

Theta series of square of Kleinian lattice Z[ (-1+sqrt(-7))/2 ].
The Gram matrix of the lattice is denoted by A in Parry 1979 on page 163.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 4*q + 12*q^2 + 16*q^3 + 28*q^4 + 24*q^5 + 48*q^6 + 4*q^7 + 60*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see p. 467, Entry 5(i).

G. C. Greubel, Table of n, a(n) for n = 0..10000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002652, A113957.

Basis( ModularForms( Gamma0(7), 2), 48) [1]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ If[ Mod[ d, 7] > 0, d, 0], {d, Divisors @ n }]]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^7] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^7])^2, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)

{a(n) = if( n<1, n==0, 4 * sigma( n / 7^valuation( n, 7)))}; /* Michael Somos, Oct 07 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 0, 0; 1, 4, 0, 0; 0, 0, 2 ,1 ; 0, 0, 1, 4], n, 1)[n])}; /* Michael Somos, Oct 07 2005 */

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, d * kronecker( 49, d)))}; /* Michael Somos, Mar 22 2012 */

ModularForms( Gamma0(7), 2, prec=48).0; # Michael Somos, Jun 12 2014

Expansion of (phi(q) * phi(q^7) + 4 * q^2 * psi(q^2) * psi(q^14))^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jul 21 2012
Expansion of (7 * P(q^7) - P(q)) / 6 where P() is a Ramanujan Eisenstein Series. - Michael Somos, Mar 22 2012
a(n) = 4 * b(n) where b(n) is multiplicative with b(p^e) = 1, if p=7, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 22 2012
G.f.: (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2.
G.f.: 1 + 4 * (Sum_{k>0} Kronecker( 49, k) * k * x^k / (1 - x^k)). - Michael Somos, Mar 22 2012
G.f.: 1 + 4 * (Sum_{k>0} x^k / (1 - x^k)^2 - 7 * x^(7*k) / (1 - x^(7*k))^2). - Michael Somos, Mar 22 2012
Convolution square of A002652. a(n) = 4 * A113957(n) unless n=0. - Michael Somos, Jul 21 2012

A318984 Theta series of quadratic form x^2 + xy + 17y^2.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Sep 06 2018

Number of integer solutions (x, y) to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)].

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 4*x^17 + 4*x^19 + 4*x^23 + 2*x^25 + 4*x^29 + 2*x^36 + 4*x^37 + 4*x^47 + 2*x^49 + 4*x^59 + 2*x^64 + 2*x^67 + 4*x^68 + 4*x^71 + 4*x^73 + 4*x^76 + ...

Jianing Song, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318982.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), this sequence (d=-67), A318985 (d=-163).

Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];
2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *)

a(n) = if(n, 2*sumdiv(n, d, kronecker(-67, d)), 1)

G.f.: 1 + 2 * Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).
a(n) = 2 * A318982(n) unless n = 0.
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(67^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, b(p^e) = e + 1 if Kronecker(-67, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(67) = 0.767613... . - Amiram Eldar, Dec 16 2023

A318985 Theta series of quadratic form x^2 + xy + 41y^2.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Sep 06 2018

Number of integer solutions (x, y) to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)].

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 4*x^41 + 4*x^43 + 4*x^47 + 2*x^49 + 4*x^53 + 4*x^61 + 2*x^64 + 4*x^71 + ...

Jianing Song, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318983.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), A318984 (d=-67), this sequence (d=-163).

Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-163, #] &]];
2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *)

a(n) = if(n, 2*sumdiv(n, d, kronecker(-163, d)), 1)

G.f.: 1 + 2 * Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
a(n) = 2 * A318983(n) unless n = 0.
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(163^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, b(p^e) = e + 1 if Kronecker(-163, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(163) = 0.49213705... . - Amiram Eldar, Dec 16 2023

A002653 Expansion of (theta_3(z)theta_3(7z)+theta_2(z)theta_2(7z))^3.

Original entry on oeis.org

1, 6, 24, 56, 114, 168, 280, 294, 444, 390, 840, 636, 1176, 1176, 1512, 1008, 1782, 2016, 1896, 2520, 3528, 2408, 3216, 2796, 4760, 3174, 5880, 4592, 6258, 4380, 5040, 6720, 7200, 6832, 10080, 7224, 8082, 7164, 12600, 7056, 14280, 11760, 12040, 9756

Offset: 0

N. J. A. Sloane

Theta series of Kleinian lattice (Z[ (-1+sqrt(-7))/2 ])^3 in 3 complex (or 6 real) dimensions.

			G.f. = 1 + 6*q + 24*q^2 + 56*q^3 + 114*q^4 + 168*q^5 + 280*q^6 + 294*q^7 + ...

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53.

Robert Israel, Table of n, a(n) for n = 0..10000
H. H. Chan and Y. L. Ong, On Eisenstein series and Sum_{m,n} q^(m^2+mn+2n^2), Proc. Amer. Math. Soc. 127 (1999), no. 6, 1735-1744, See page 1737. MR1600120 (99i:11029).
N. Elkies, The Klein quartic in number theory

Cf. A002652.

A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 6*A[2] + 24*A[3] + 56*A[4] +114*A[5] + 168*A[6] + 280*A[7];  /* Michael Somos, Nov 09 2014 */

g:= (JacobiTheta3(0,z) * JacobiTheta3(0,z^7) + JacobiTheta2(0,z) * JacobiTheta2(0,z^7))^3:
S:= series(g,z,101):
seq(coeff(S,z,i),i=0..100); # Robert Israel, Aug 12 2020

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^7] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^7])^3, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

{a(n) = local(A, t2, t3); if( n<1, n==0, A = x * O(x^n); t2 = sum(k=1, (sqrtint(4*n + 1) + 1)\2, 2*x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2*x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3, n))}; /* Michael Somos, Jun 03 2005 */

{a(n) = local(A, t1, t7); if( n<0, 0, A = x * O(x^n); t1 = eta(x + A)^4; t7 = eta(x^7 + A)^4; polcoeff( (t1^2 + 13 * x * t1 * t7 + 49 * x^2 * t7^2) / (t1 * t7)^(1/4), n))}; /* Michael Somos, Mar 11 2008 */

A = ModularForms( Gamma1(7), 3, prec=60) . basis(); (3*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # _Michael Somos, May 25 2014

G.f.: (theta_3(z)*theta_3(7*z) + theta_2(z)*theta_2(7*z))^3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - Michael Somos, Jun 03 2005
Expansion of (eta(q)^8 + 13 * eta(q)^4 * eta(q^7)^4 + 49 * eta(q^7)^8) / ( eta(q) * eta(q^7) ) in power of q. - Michael Somos, Mar 11 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (7*t)) = 7^(3/2)*(t/i)^3*f(t) where q = exp(2*Pi*i*t).

A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 25 2007, Oct 04 2008

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

G.f. is called omega(q) by Berkovich and Yesilyurt.

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^11 + x^12 + 2*x^14 + 2*x^18 + 2*x^21 + x^24 + ...
G.f. = q + q^7 + q^9 + 2*q^11 + 2*q^23 + q^25 + 2*q^29 + 2*q^37 + 2*q^43 + q^49 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Alexander Berkovich and Hamza Yesilyurt, New Identities For 7-cores with Prescribed BG-Rank, arXiv:math/0603150 [math.NT], 2006-2007, page 3, equation (1.19).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002652, A033719, A035162, A035182, A110399, A121454.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[#, 2] KroneckerSymbol[ -28, #] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(7/2)], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (d%2) * kronecker( -28, 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 == 7, 1, 1 == kronecker( -7, p), e + 1, 1-e%2)))};

Expansion of psi(x^4) * phi(x^14) + x^3 * psi(x^28) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(7^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), b(p^e) = e + 1 if p == 1, 2, 4 (mod 7).
a(7*n + 1) = a(7*n + 2) = a(7*n + 6) = 0. a(7*n + 3) = a(n).
Expansion of psi(q) * psi(q^7) - q * psi(q^2) * psi(q^14) = (psi(q) * psi(q^7) + psi(-q) * psi(-q^7)) / 2 in powers of q^2 where psi() is a Ramanujan theta function.
a(n) = A035162(2*n + 1) = A035182(2*n + 1) = A110399(2*n + 1) = A121454(2*n + 1).
2 * a(n) = A002652(2*n + 1) = A033719(2*n + 1). - Michael Somos, Dec 30 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Dec 29 2023

A279618 Expansion of w_7/(1 + 13w_7 + 49w_7^2) in powers of q, where w_7 = (eta(7*q)/eta(q))^4.

Original entry on oeis.org

1, -9, 30, -15, -240, 978, -1463, -2361, 18201, -42800, 15624, 227742, -809028, 1088367, 1593120, -11383551, 25003158, -8589729, -119069358, 403991280, -521730930, -736063496, 5088063696, -10843708302, 3624181875, 48991048836, -162420646812, 205328313785, 284014016994

Offset: 1

Lynette O'Brien, Dec 15 2016

sign

G.f. is  y_7 in Cooper's paper.
See Equation (3.15) and Theorem 3.10 in O'Brien's thesis.
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 07 2018

			G.f. = q - 9*q^2 + 30*q^3 - 15*q^4 - 240*q^5 + 978*q^6 - 1463*q^7 + ...

S. Cooper, (2012). Sporadic sequences, modular forms and new series for 1/pi. The Ramanujan Journal, 29(1-3), 163-183.
L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

Lynette O'Brien, Modular forms and two new integer sequences at level 7
L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

Cf. A002652, A121593, A279613, A279619.

a[ n_] := With[{u1 = QPochhammer[ x]^4, u7 = QPochhammer[ x^7]^4}, SeriesCoefficient[ x u1 u7 / (u1^2 + 13 x u1 u7 + 49 x^2 u7^2) , {x, 0, n}]]; (* Michael Somos, Sep 07 2018 *)

{a(n) = my(A); if( n<1, 0, A = x * O(x^n); A = x * (eta(x^7 + A) / eta(x + A))^4; polcoeff( 1 / (1/A + 13 + 49*A), n))}; /* Michael Somos, Sep 07 2018 */

G.f. is w_7/(1 + 13*w_7 + 49*w_7^2) = (eta(q)*eta(7q)/z_7)^3 where w_7 = (eta(7*q)/eta(q))^4 and z_7 = 1 + 2*Sum_{k>0} Kronecker(-7,k)*q^k/(1-q^k).

G.f. is also (eta(q)*eta(7*q)/z_7)^3, where z_7 = 1 + 2*Sum_{k>0} Kronecker(-7,k)*q^k/(1-q^k). See A002652.

A028596 Expansion of (theta_3(z)theta_3(7z) + theta_2(z)theta_2(7z))^4.

Original entry on oeis.org

1, 8, 40, 128, 328, 656, 1216, 1864, 2856, 3560, 5392, 6368, 9856, 10640, 17000, 16832, 22600, 23760, 32776, 32576, 43792, 52864, 57568, 58560, 78528, 76024, 94864, 98432, 137864, 116720, 152512, 143040, 179240, 179072, 212112, 237328, 265768, 242352, 296704, 295232

Offset: 0

N. J. A. Sloane

nonn

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A002652, A002653, A028594.

a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 4], n, 1)))^4, n); \\ Jinyuan Wang, Feb 21 2020

More terms from Jinyuan Wang, Feb 21 2020

A058565 McKay-Thompson series of class 21C for the Monster group.

Original entry on oeis.org

1, 3, 8, 11, 25, 35, 57, 86, 139, 198, 291, 417, 588, 812, 1132, 1538, 2103, 2805, 3767, 4963, 6554, 8548, 11165, 14426, 18601, 23830, 30443, 38642, 48986, 61748, 77669, 97206, 121478, 151067, 187556, 231974, 286385, 352340, 432641, 529688, 647241, 788738, 959470, 1164291, 1410386

Offset: 0

N. J. A. Sloane, Nov 27 2000

nonn

			G.f. = 1 + 3*x + 8*x^2 + 11*x^3 + 25*x^4 + 35*x^5 + 57*x^6 + 86*x^7 + ... -  _Michael Somos_, Feb 26 2017
T21C = 1/q + 3*q^2 + 8*q^5 + 11*q^8 + 25*q^11 + 35*q^14 + 57*q^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 176 Entry 32(iii).

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A002652, A002655, A282877.

a[ n_] := With[ {A = (QPochhammer[ x^7] / QPochhammer[ x])^4}, SeriesCoefficient[ (1/A + 13 x + 49 x^2 A)^(1/3), {x, 0, n}]]; (*  Michael Somos, Feb 26 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(1/3)*(eta[q]*eta[q^7]/(eta[q^2] *eta[q^14])); a:= CoefficientList[Series[(A + 4*q/A^2), {q,0,60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 21 2018 *)
a[ n_] := With[ {A1 = QPochhammer[ x] QPochhammer[ x^7], A2 = QPochhammer[ x^2] QPochhammer[ x^14]}, SeriesCoefficient[ (A1^3 + 4 x A2^3) / (A1^2 A2), {x, 0, n}]]; (*  Michael Somos, Oct 27 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1/A + 13*x + 49*x^2 * A)^(1/3), n))}; /*  Michael Somos, Feb 26 2017 */

q='q+O('q^50); A = (eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))); Vec(A + 4*q/A^2) \\ G. C. Greubel, Jun 21 2018

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1^2 * A2), n))}; /* Michael Somos, Oct 27 2018 */

From Michael Somos, Feb 26 2017: (Start)
Expansion of f(-x^7, -x^14)^2 / f(-x, -x^2) * (w3/w1^2 + x*w2/w3^2 - x*w1/w2^2) in powers of x where w1 = f(-x, -x^6), w2 = f(-x^2, -x^5), w3 = f(-x^3, -x^4) and f(, ) is Ramanujan's general theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = f(t) where q = exp(2 Pi i t).
Convolution cube is A282877.
Convolution product with A002655 is A002652. (End)
Expansion of A + 4*q/A^2, where A = q^(1/3)*(eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))), in powers of q. - G. C. Greubel, Jun 21 2018
a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Feb 26 2017

Terms a(8) onward added by G. C. Greubel, Jun 21 2018

A129666 Expansion of unique cusp form of weight 4 level 7 in powers of q.

Original entry on oeis.org

1, -1, -2, -7, 16, 2, -7, 15, -23, -16, -8, 14, 28, 7, -32, 41, 54, 23, -110, -112, 14, 8, 48, -30, 131, -28, 100, 49, -110, 32, 12, -161, 16, -54, -112, 161, -246, 110, -56, 240, 182, -14, 128, 56, -368, -48, 324, -82, 49, -131, -108, -196, -162, -100, -128

Offset: 1

Michael Somos, Apr 27 2007

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

			G.f. = q - q^2 - 2*q^3 - 7*q^4 + 16*q^5 + 2*q^6 - 7*q^7 + 15*q^8 - 23*q^9 - ...

H. Rosson and G. Tornaria, Central values of quadratic twists for a modular form of weight 4, pp. 315-321 of J. B. Conrey et al., ed., Ranks of Elliptic Curves and Random Matrix Theory, Cambridge University Press, 2007.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006, pp. 107-115. See c7() p. 108.
LMFDB, Space of modular forms of level 7 and weight 4.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002652, A002656.

Basis( CuspForms( Gamma0(7), 4), 56)[1]; /* Michael Somos, Nov 11 2015 */

a[ n_] := With[ {A1 = QPochhammer[ q] QPochhammer[ q^7], A2 = QPochhammer[ q^2] QPochhammer[ q^14]}, SeriesCoefficient[  (A1^3 + 4 q A2^3) A1^2 / A2, {q, 0, n}]]; (* Michael Somos, Nov 11 2015 *)

{a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4*x * A2^3) * A1^2 / A2, n))};

CuspForms( Gamma0(7), 4, prec=55).0; # Michael Somos, May 28 2013

Expansion of q * phi(-q)^3 * psi(q) * phi(-q^7)^3 * psi(q^7) + 4*q^2 * (phi(-q) * psi(q) * phi(-q^7) * psi(q^7))^2 in powers of q.
Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) * (eta(q) * eta(q^7))^2 / (eta(q^2) * eta(q^14)) in powers of q.
a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 49 (t/i)^4 f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 + 2*u*v + 16*u*w + 12*v^2 + 32*v*w + 256*w^2) * (-v^3 + 2*w*u*v + w*u^2 + 16*w^2*u) + 2*v^5.
Convolution of A002652 and A002656.

cp's OEIS Frontend

A138811 Theta series of quadratic form x^2 + x*y + 11*y^2.

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A028594 Expansion of (theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7))^2 in powers of q.

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A318984 Theta series of quadratic form x^2 + x*y + 17*y^2.

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A318985 Theta series of quadratic form x^2 + x*y + 41*y^2.

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A002653 Expansion of (theta_3(z)*theta_3(7z)+theta_2(z)*theta_2(7z))^3.

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A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.

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A138811 Theta series of quadratic form x^2 + xy + 11y^2.

A318984 Theta series of quadratic form x^2 + xy + 17y^2.

A318985 Theta series of quadratic form x^2 + xy + 41y^2.

A002653 Expansion of (theta_3(z)theta_3(7z)+theta_2(z)theta_2(7z))^3.

A279618 Expansion of w_7/(1 + 13w_7 + 49w_7^2) in powers of q, where w_7 = (eta(7*q)/eta(q))^4.

A028596 Expansion of (theta_3(z)theta_3(7z) + theta_2(z)theta_2(7z))^4.