A121373 -id:A121373 - cp's OEIS frontend

A112605 Number of representations of n as a sum of a square and six times a triangular number.

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

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
G.f. = 1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - _Michael Somos_, Aug 11 2009
G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - Michael Somos, Aug 11 2009

a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* Michael Somos, May 20 2006 */

{a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* Michael Somos, May 20 2006 */

a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos, May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos, May 20 2006
a(n)=A002324(4n+3). - Michael Somos, May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, May 20 2006, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164273. - Michael Somos, Aug 11 2009
a(3*n + 2) = 0. - Michael Somos, Aug 11 2009

A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.

Original entry on oeis.org

1, 1, -1, 0, 1, 0, -1, -1, 2, 1, -2, -1, 2, 1, -3, -1, 4, 2, -5, -2, 5, 2, -6, -3, 8, 4, -9, -4, 10, 4, -12, -6, 15, 7, -17, -7, 19, 8, -22, -10, 26, 12, -30, -13, 33, 14, -38, -17, 45, 21, -51, -22, 56, 24, -64, -29, 74, 33, -83, -36, 92, 40, -104, -46, 119, 53, -133, -58, 147, 63, -165, -73, 187, 83, -208, -90, 229, 99, -256

Offset: 0

N. J. A. Sloane

sign

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

			G.f. = 1 + x - x^2 + x^4 - x^6 - x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + 2*x^12 + ...
G.f. = 1/q + q^7 - q^15 + q^31 - q^47 - q^55 + 2*q^63 + q^71 - 2*q^79 - q^87 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A029839, A082303, A083365.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ q^2, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q] / QPochhammer[ q^4, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ q^(1/8) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
(QPochhammer[-x, x^2, 1/2] + O[x]^100)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 + x^k)^(-(-1)^k), 1 + x * O(x^n)), n))};

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[1,1] / A[2,1] + x * O(x^n), n))}; /* Michael Somos, Mar 02 2006 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = subst(A, x, x^2); A = sqrt((A2 + 2  * x / A2) / A)); polcoeff(A, n))};

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

Expansion of f(x) / f(-x^4) = phi(x) / psi(x) = psi(x) / psi(x^2) = phi(-x^2) / psi(-x) = chi(x) * chi(-x^2) = chi^2(x) * chi(-x) = chi^2(-x^2) / chi(-x) = (phi(x) / psi(x^2))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/8) * eta(q^2)^3 / (eta(q) * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ 1, -2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^4 - u^4*v^2. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^4 - v^4 - 4*u*v + u^3*v^3. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + w^2 - u^2*v*w. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2^2 + u6^2 - u1*u2*u3*u6. - Michael Somos, Mar 02 2006
G.f. A(x) satisfies A(x)^2 = (A(x^4) + 2*x / A(x^4)) / A(x^2). - Michael Somos, Mar 08 2004
G.f. A(x) satisfies A(x) = (A(x^2)^2+4*x/A(x^2)^2)^(1/4). - Joerg Arndt, Aug 06 2011
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 + x^(2*k)) = (Sum_{k>0} x^((k^2 - k)/2)) / (Sum_{k>0} x^(k^2 - k)).
G.f.: 1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...))).
A082303(n) = (-1)^n a(n). Convolution square is A029839. Convolution inverse is A083365.
G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
G.f.: (-x; x^2){1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

A081360 Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

Original entry on oeis.org

1, -1, 1, -2, 2, -3, 4, -5, 6, -8, 10, -12, 15, -18, 22, -27, 32, -38, 46, -54, 64, -76, 89, -104, 122, -142, 165, -192, 222, -256, 296, -340, 390, -448, 512, -585, 668, -760, 864, -982, 1113, -1260, 1426, -1610, 1816, -2048, 2304, -2590, 2910, -3264, 3658, -4097, 4582, -5120, 5718, -6378

Offset: 0

Michael Somos, Mar 18 2003

sign

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

Number of partitions of n into distinct parts with an even number of odd parts minus partitions of n into distinct parts with an odd number of odd parts. G.f.: Product_{i=1..oo} (1+(-1)^i*x^i). - Jon Perry, Jun 04 2004

			G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 5*x^7 + 6*x^8 - 8*x^9 + ...
G.f. = q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=0. - _N. J. A. Sloane_, Aug 31 2014
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
E. W. Weisstein's World of Mathematics, Elliptic Lambda Function
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A000009, A000700, A278428.

read theta; t1:=series(eta,q,48); t2:= q^(-1/24)*t1*subs(q=q^4,t1)/subs(q=q^2,t1)^2; series(t2,q,48); seriestolist(%); # N. J. A. Sloane, Aug 24 2007

a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ With[ {m = ModularLambda[ Log[ q] / (Pi I)]}, ( m (1 - m) / (16 q))^(1/24)], {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(QPochhammer[-1, -x]/2 + O[x]^60)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

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

Expansion of 1 / chi(x) = chi(-x) / chi(-x^2) = f(x) / phi(x) = f(-x) / phi(-x^2) = psi(-x) / f(-x^2) = f(-x^2) / f(x) = f(-x^4) / psi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (lambda * (1 - lambda) / (16 * q))^(1/24) in powers of q where lambda is a modular elliptic function and q = exp(Pi i z) is the nome. - Michael Somos, Jul 19 2012
Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.
Expansion of q^(-1/24) / f(t) in powers of q = exp(Pi i t) where f() is Weber's function.
Euler transform of period 4 sequence [-1, 1, -1, 0, ...].
Given g.f. A(x), B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u - v^2) * (v - u^2) - (4 * u * v * (1 - u*v))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 16 2007
G.f.: Product_{k>0} 1 / ( 1 + x^(2k - 1)) = Product_{k>0} (1 + (-x)^k).
a(n) = (-1)^n * A000009(n). Convolution inverse of A000700.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
G.f.: (1/2)*(-1; -x)inf, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 21 2016
G.f.: exp(-Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
Given g.f. A(x), B(x) = 2^(1/4) * x * A(x^24) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^6 + v^6 + 2*u*v * ((u*v)^4 - 1). - Michael Somos, Mar 14 2019

A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

Original entry on oeis.org

1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624

Offset: 0

N. J. A. Sloane

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

k^2 is the parameter and q the Jacobi nome of elliptic functions. See, e.g., Fricke, p. 11, eq. (8) with p. 10. eq. (1). - Wolfdieter Lang, Jul 04 2016

			G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...
G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A000700, A001936, A010054, A079006, A093160, A121373, A127393, A127931, A127932, A139820.

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)

{a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( mMichael Somos, Mar 26 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^4, n))}; /* Michael Somos, Mar 26 2004 */

Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. - Joerg Arndt, Aug 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
a(n) = (-1)^n * A093160(n). Convolution square of A079006.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Edited by N. J. A. Sloane, Mar 31 2007

A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1. - Jon Perry, Apr 04 2004
Also, number of partitions of n into parts of -2 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 68 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q - 2*q^13 - q^25 + 2*q^37 + q^49 + 2*q^61 - 2*q^73 - 2*q^97 - 2*q^109 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
G. E. Andrews, Advanced problems 6562, Amer. Math. Monthly 94, 1987.
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
S. Cooper, M. D. Hirschhorn and R. Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
J. T. Joichi, Hecke-Rogers, Andrews identities; combinatorial proofs, Discrete Mathematics, Vol. 84, Issue 3, 1990, pp. 255-259.
Victor G. Kac and Dale H. Peterson, Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms, Advances in Mathematics (1984), 53. 125-264, see page 261, (5.19).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
V. Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000712 (reciprocal of g.f.), A010815, A010816, A258406.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A010815 - A010840.

# DedekindEta is defined in A000594.
A002107List(len) = DedekindEta(len, 2)
A002107List(78) |> println # Peter Luschny, Mar 09 2018

Basis( CuspForms( Gamma1(144), 1), 926) [1]; /* Michael Somos, May 17 2015 */

A010816 := proc (n); if frac(sqrt(8*n+1)) = 0 then (-1)^((1/2)*isqrt(8*n+1)-1/2)*isqrt(8*n+1) else 0 end if; end proc:
N := 10:
a := proc (n) option remember; if n < 0 then 0 else A010816(n) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = -N..-1) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = 1..N) end if; end proc:
seq(a(n), n = 0..100); # Peter Bala, Apr 06 2022

terms = 78; Clear[s]; s[n_] := s[n] = Product[(1 - x^k)^2, {k, 1, n}] // Expand // CoefficientList[#, x]& // Take[#, terms]&; s[n = 10]; s[n = 2*n]; While[s[n] != s[n - 1], n = 2*n]; A002107 = s[n] (* Jean-François Alcover, Jan 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = my(A, p, e, x); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^((p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare(p - 9*i^2), x=i; break)); (e + 1) * (-1)^(e*x))))}; /* Michael Somos, Aug 30 2006 */

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^2, n))}; /* Michael Somos, Aug 30 2006 */

Vec(eta(x)^2) \\ Charles R Greathouse IV, Apr 22 2016

Expansion of q^(-1/12) * eta(q)^2 in powers of q. - Michael Somos, Mar 06 2012
Euler transform of period 1 sequence [ -2, ...]. - Michael Somos, Mar 06 2012
a(n) = b(12*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e*x) if p == 1 (mod 12) where p = x^2 + 9*y^2. - Michael Somos, Sep 16 2006
Convolution inverse of A000712.
a(0) = 1, a(n) = -(2/n)*Sum{k = 0..n-1} a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
Expansion of f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, May 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 17 2015
a(n) = Sum_{k=0..n} A010815(k)*A010815(n-k); self convolution of A010815. - Gevorg Hmayakyan, Sep 18 2016
G.f.: Sum_{m, n in Z, n >= 2*|m|} (-1)^n * x^((3*(2*n + 1)^2 - (6*m + 1)^2)/24). - Seiichi Manyama, Oct 01 2016
G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
From Peter Bala, Jan 02 2021: (Start)
For prime p congruent to 5, 7 or 11 (mod 12), a(n*p^2 + (p^2 - 1)/12) = e*a(n), where e = 1 if p == 7 or 11 (mod 12) and e = -1 if p == 5 (mod 12).
If n and p are coprime then a(n*p + (p^2 - 1)/12) = 0. See Cooper et al., Theorem 1. (End)
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = A010816(n) + Sum_{k a nonzero integer} (-1)^(k+1)*a(n - k*(3*k-1)/2), where A010816(n) = (-1)^m*(2*m+1) if n = m*(m + 1)/2, with m positive, is a triangular number else equals 0. For example, n = 10 = (4*5)/2 is a triangular number, A010816(10) = 9, and so a(10) = 9 + a(9) + a(8) - a(5) - a(3) = 9 - 2 - 2 - 2 - 2 = 1. - Peter Bala, Apr 06 2022

A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of representations of n as the sum of a triangular number and twice a triangular number. - James Sellers, Dec 21 2005
Also the number of positive odd solutions to equation x^2 + 2*y^2 = 8*n + 3. - Seiichi Manyama, May 28 2017

			G.f. = 1 + x + x^2 + 2*x^3 + x^5 + 2*x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^12 + ...
G.f. = q^3 + q^11 + q^19 + 2*q^27 + q^43 + 2*q^51 + q^59 + q^67 + q^75 + q^83 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
Iana I. Anguelova, The two bosonizations of the CKP hierarchy: overview and character identities, arXiv:1708.04992 [math-ph], 2017.
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A027414, A097723, A033761-A033807, A082564.

A := Basis( ModularForms( Gamma1(32), 1), 840); A[4] + A[12]; /* Michael Somos, Jan 31 2015 */

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A002325 := proc(n) sigmamr(n,8,1)+sigmamr(n,8,3)-sigmamr(n,8,5)-sigmamr(n,8,7) ; end proc:
A033761 := proc(n) A002325(8*n+3)/2 ; end proc:
seq(A033761(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] / 4, {q, 0, 2 n + 3/4}]; (* Michael Somos, Nov 16 2011 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^4]^2/QP[q]) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A), n))}; /* Michael Somos, Jul 05 2006 */

Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of psi(q) * psi(q^2) in powers of q where psi() is a Ramanujan theta function.
Expansion of q^(-3/8) * eta(q^2) * eta^2(q^4) / eta(q) in powers of q. - Michael Somos, Jul 05 2006
Expansion of q^(-3/4) * (theta_2(q) * theta_2(q^2)) / 4 in powers of q^2. - Michael Somos, Jul 05 2006
Given g.f. A(x), then B(x) = x^3 * A(x^8) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^2 + 3*u2^2*u3^4 - 4*u1*u2*u3*u6 * (u2^2 + 3*u6^2). - Michael Somos, Jul 05 2006
a(n) = A002325(8*n+3)/2. [Hirschhorn] - R. J. Mathar, Mar 23 2011
a(n) = A027414(8*n + 3). - Michael Somos, Nov 16 2011
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082564. - Michael Somos, Jan 31 2015
From Peter Bala, Jan 07 2021: (Start)
G.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(8*n + 3)). See Agarwal, p. 285, equation 6.19.
A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(8*n + 3)). Cf. A121444. (End)
A(q^2) = (1/2)*Sum_{k >= 0} q^k/(1 + q^(4*k+3)) + (1/2)*Sum_{k >= 0} q^(3*k)/(1 + q^(4*k+1)) - set z = 1 and replace q with q^2 in Anguelova, equation 3.35. - Peter Bala, Mar 03 2021

More terms from Vladeta Jovovic, Sep 14 2004

A089801 a(n) = 0 unless n = 3j^2 + 2j or 3j^2 + 4j + 1 for some j >= 0, in which case a(n) = 1.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also characteristic function of generalized octagonal numbers A001082. - Omar E. Pol, Jul 13 2012
Number 12 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^56 + x^65 + x^85 + ...
G.f. = q + q^4 + q^16 + q^25 + q^49 + q^64 + q^100 + q^121 + q^169 + q^196 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Th. 2.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See X(q).
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
Index entries for characteristic functions.

Cf. A001082, A033684, A080995, A089802, A089807, A280739.

Basis( ModularForms( Gamma0(36), 1/2), 87) [2]; /* Michael Somos, Jul 02 2014 */

A089801 := proc(n)
        A033684(3*n+1) ;
end proc: # R. J. Mathar, Oct 07 2011
M:=33;
S:=f->series(f,q,500);
L:=f->seriestolist(f);
X:=add(q^(3*n^2+2*n),n=-M..M);
S(%);
L(%); # N. J. A. Sloane, Jan 31 2012
eps:=Array(0..120,0);
for j from 0 to 120 do
if 3*j^2+2*j <= 120 then eps[3*j^2+2*j] := 1; fi;
if 3*j^2+4*j+1 <= 120 then eps[3*j^2+4*j+1] := 1; fi;
end do;  # N. J. A. Sloane, Aug 12 2017

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/3) (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3]), {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Jun 29 2012 *)

{a(n) = issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */

from sympy.ntheory.primetest import is_square
def A089801(n): return int(is_square(3*n+1)) # Chai Wah Wu, May 17 2023

G.f.: Sum_{n=-oo..oo} q^(3n^2+2n).
Expansion of Jacobi theta function (theta_3(q^(1/3)) - theta_3(q^3))/(2 q^(1/3)) in powers of q.
Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -1, ...]. - Michael Somos, Apr 12 2005
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p != 3. - Michael Somos, Jun 06 2005; b=A033684. - R. J. Mathar, Oct 07 2011
Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(x) * psi(-x^3) in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 19 2007
Expansion of f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089807.
a(8*n + 4) = a(4*n + 2) = a(4*n + 3) = 0, a(4*n + 1) = a(n), a(8*n) = A080995(n). - Michael Somos, Jan 28 2011
a(n) = (-1)^n * A089802(n).
For n > 0, a(n) = b(n)-b(n-1) + c(n)-c(n-1), where b(n) = floor(sqrt(n/3+1/9)+2/3) and c(n) = floor(sqrt(n/3+1/9)+4/3). - Mikael Aaltonen, Jan 22 2015
a(n) = A033684(3*n + 1). - Michael Somos, Jan 10 2017

Edited with simpler definition by N. J. A. Sloane, Jan 31 2012

Further edited by N. J. A. Sloane, Aug 12 2017

A115977 Expansion of elliptic modular function lambda in powers of the nome q.

Original entry on oeis.org

16, -128, 704, -3072, 11488, -38400, 117632, -335872, 904784, -2320128, 5702208, -13504512, 30952544, -68901888, 149403264, -316342272, 655445792, -1331327616, 2655115712, -5206288384, 10049485312, -19115905536, 35867019904, -66437873664

Offset: 1

Michael Somos, Feb 09 2006

sign

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

			G.f. = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Dieckmann, Collection of Infinite Products and Series
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
Wolfram Research Basic Algebraic Identities Relations involving squares, 1st formula

Cf. A005798, A014972, A128692, A132136.

a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/16 (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q^2])^8, {q, 0, n}]; (* Michael Somos, May 26 2016 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))};

Expansion of Jacobi elliptic parameter m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.
Expansion of 16 * q * (psi(q^2) / phi(q))^4 = 16 * q * (psi(q^2) / psi(q))^8 = 16 * q * (psi(q) / phi(q))^8 = 16 * q * (psi(-q) / phi(-q^2))^8 = 16 * q / (chi(q) * chi(-q^2))^8 = 16 * q * (f(-q^4) / f(q))^8  in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - v)^2 - 16 * v * (1 - u).
lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692.
G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.
a(n) = 16 * A005798(n). a(n) = -(-1)^n * A014972(n) unless n=0.
a(n) = -(-1)^n * A132136(n). - Michael Somos, Jun 03 2015
Empirical: Sum_{n>=1}(exp(-2*Pi)^n*a(n)) = 17 - 12*sqrt(2). - Simon Plouffe, Feb 20 2011
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Apr 06 2018
The g.f. A(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... satisfies A(q) + A(-q) = A(q)*A(-q). - Peter Bala, Sep 26 2023

A000143 Number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

			1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 314 - 315.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

8th column of A286815. - Seiichi Manyama, May 27 2017
Row d=8 of A122141.
Cf. A008457, A035016, A010815.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

# JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

(sum(x^(m^2),m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - Vaclav Kotesovec, Jul 12 2024

A002652 Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

Original entry on oeis.org

1, 2, 4, 0, 6, 0, 0, 2, 8, 2, 0, 4, 0, 0, 4, 0, 10, 0, 4, 0, 0, 0, 8, 4, 0, 2, 0, 0, 6, 4, 0, 0, 12, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 12, 0, 8, 0, 0, 2, 4, 0, 0, 4, 0, 0, 8, 0, 8, 0, 0, 0, 0, 2, 14, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0, 4, 4, 6, 0

Offset: 0

N. J. A. Sloane

In other words, theta series of lattice with Gram matrix [2, 1; 1, 4].
The number of integer solutions (x, y) to x^2 + x*y + 2*y^2 = n.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^2 + 6*x^4 + 2*x^7 + 8*x^8 + 2*x^9 + 4*x^11 + 4*x^14 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 4] = 1 + 2*q^2 + 4*q^4 + 6*q^8 + 2*q^14 + 8*q^16 + 2*q^18 + 4*q^22 + 4*q^28 + 10*q^32 + 4*q^36 + 8*q^44 + 4*q^46 + 2*q^50 + 6*q^56 + 4*q^58 + 12*q^64 + 6*q^72 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 467, Entry 5(i).

John Cannon, Table of n, a(n) for n = 0..5000
J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035182, A133827, A028951.
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), this sequence (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(14), 1), 85); A[1] + 2*A[2] + 4*A[3] + 6*A[5]; /* Michael Somos, Jun 10 2015 */

f[d_] := KroneckerSymbol[-7, d]; a[n_] := 2*Total[f /@ Divisors[n]]; a[0]=1; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(t2, t3); if( n<1, n==0, t2 = 2 * sum( n=1, (sqrtint( max(0, 4*n - 7)) + 1)\2, x^(n*n - n)); t3 = 1 + 2 * sum( n=1, sqrtint(n), x^(n*n)); polcoeff( t3 * subst(t3, x, x^7) + x^2 * t2 * subst(t2, x, x^7), n))};

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n*4\7), 2 * issquare(t = 4*n - 7*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{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 * A2), n))}; /* Michael Somos, May 28 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 1; 1, 4], n, 1)[n])}; /* Michael Somos, Jun 03 2005 */

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -7, d)))}; /* Michael Somos, Oct 07 2005 */

G.f.: theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7).
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-7, k) * x^k / (1 - x^k). - Michael Somos, Mar 17 2012
Expansion of phi(x) * phi(x^7) + 4 * x^2 * psi(x^2) * psi(x^14) = phi(-x) * phi(-x^7) + 4 * x * psi(x) * psi(x^7) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 17 2012
Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) / (eta(q) * eta(q^2) * eta(q^7) * eta(q^14)) in powers of q. - Michael Somos, May 28 2005
Moebius transform is period 7 sequence [ 2, 2, -2, 2, -2, -2, 0, ...]. - Michael Somos, Oct 07 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5 * v^2 + 4 * w^2 + 2 * u*w - 4 * u*v - 8 * v*w. - Michael Somos, Sep 20 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 + 2*u2^3*u3 + 18*u1*u3*u6^2 + 18*u2*u3^2*u6 + 6*u1*u2^2*u6 + 3*u1^2*u2*u3 - 3*u2*u3^3 - 18*u2*u3*u6^2 - 6*u1*u6^3 - 9*u1*u3^2*u6 - 6*u1*u2^2*u3 - 6*u1^2*u2*u6. - Michael Somos, Jun 03 2005
From Michael Somos, Mar 17 2012: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = 2 * A035182(n) unless n = 0. a(7*n + 5) = a(7*n + 6) = a(9*n + 3) = a(9*n + 6) = 0. a(2*n + 1) = 2 * A133827(n). a(9*n) = a(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(7^e) = 1, b(p^e) = e + 1 if p == 1, 2, 4 (mod 7), b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, Jun 10 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(7) = 2.3748208... . - Amiram Eldar, Dec 16 2023

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A112605 Number of representations of n as a sum of a square and six times a triangular number.

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A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.

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A081360 Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

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A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

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A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

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Julia

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A033761 Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).

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