A104794 -id:A104794 - cp's OEIS frontend

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

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

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present sequence gives the expansion coefficients of psi(q)^2.

Also the number of positive odd solutions to equation x^2 + y^2 = 8*n + 2. - Seiichi Manyama, May 28 2017

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 3*x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^11 + ...
G.f. for B(q) = q * A(q^4) = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag. See p. 139 Example (iv).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
R. W. Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. See p. 279.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. [See Pi_q.]
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916. See vol. 2, p 31, Article 272.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 165.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. See p. 108.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp. 73-94.
Hjalmar Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).
Cf. A004020, A005883, A104794, A052343, A199015 (partial sums).
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A002175, A053692, A113407, A121444.
Cf. A000217, A001938, A004018.
Cf. A274621 (reciprocal series).
Cf. A001826, A001842.
Cf. A019669, A079006.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A := Basis( ModularForms( Gamma1(8), 1), 420); A[2]; /* 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:
A002654 := proc(n) sigmamr(n,4,1)-sigmamr(n,4,3) ; end proc:
A008441 := proc(n) A002654(4*n+1) ; end proc:
seq(A008441(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

Plus@@((-1)^(1/2 (Divisors[4#+1]-1)))& /@ Range[0, 104] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q], {q, 0, n + 1/4}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q]^2, {q, 0, 2 n + 1/2}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]];  (* Michael Somos, Jun 08 2014 *)
QP = QPochhammer; s = QP[q^2]^4/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
TriangleQ[n_] := IntegerQ@Sqrt[8n +1]; Table[Count[FrobeniusSolve[{1, 1}, n], {?TriangleQ}], {n, 0, 104}] (* Robert G. Wilson v, Apr 15 2017 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k * (k+1)/2), x * O(x^n))^2, n) )};

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, (-1)^(d\2)))}; /* Michael Somos, Sep 02 2005 */

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

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 14 2005 */

{ my(q='q+O('q^166)); Vec(eta(q^2)^4 / eta(q)^2) } \\ Joerg Arndt, Apr 16 2017

ModularForms( Gamma1(8), 1, prec=420).1; # Michael Somos, Jun 08 2014

This sequence is the quadrisection of many sequences. Here are two examples:
a(n) = A002654(4n+1), the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002
a(n) = b(4*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Sep 14 2005
G.f.: (Sum_{k>=0} x^((k^2 + k)/2))^2 = (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).
Expansion of Jacobi theta (theta_2(0, sqrt(q)))^2 / (4 * q^(1/4)).
Sum[d|(4n+1), (-1)^((d-1)/2) ].
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4 * v * w^2 - u^2 * w. - Michael Somos, Sep 14 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1 * u3 - (u2 - u6) * (u2 + 3*u6). - Michael Somos, Sep 14 2005
Expansion of Jacobi k/(4*q^(1/2)) * (2/Pi)* K(k) in powers of q^2. - Michael Somos, Sep 14 2005. Convolution of A001938 and A004018. This appears in the denominator of the Jacobi sn and cn formula given in the Abramowitz-Stegun reference, p. 575, 16.23.1 and 16.23.2, where m=k^2. - Wolfdieter Lang, Jul 05 2016
G.f.: Sum_{k>=0} a(k) * x^(2*k) = Sum_{k>=0} x^k / (1 + x^(2*k + 1)).
G.f.: Sum_{k in Z} x^k / (1 - x^(4*k + 1)). - Michael Somos, Nov 03 2005
Expansion of psi(x)^2 = phi(x) * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Jan 25 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A104794.
Euler transform of period 2 sequence [ 2, -2, ...].
G.f.: q^(-1/4) * eta(q^2)^4 / eta(q)^2. See also the Fine reference.
a(n) = Sum_{k=0..n} A010054(k)*A010054(n-k). - Reinhard Zumkeller, Nov 03 2009
A004020(n) = 2 * a(n). A005883(n) = 4 * a(n).
Convolution square of A010054.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^(2*k-1))^2.
a(2*n) = A113407(n). a(2*n + 1) = A053692(n). a(3*n) = A002175(n). a(3*n + 1) = 2 * A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jun 08 2014
G.f.: exp( Sum_{n>=1} 2*(x^n/n) / (1 + x^n) ). - Paul D. Hanna, Mar 01 2016
a(n) = A001826(2+8*n) - A001842(2+8*n), the difference between the number of divisors 1 (mod 4) and 3 (mod 4) of 2+8*n. See the Ono et al. link, Corollary 1, or directly the Niven et al. reference, p. 165, Corollary (3.23). - Wolfdieter Lang, Jan 11 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 - x^1)^2 / (1 - x^3 + x^2*(1 - x^2)^2 / (1 - x^5 + x^3*(1 - x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
Given g.f. A(x), and B(x) is the g.f. for A079006, then B(x) = A(x^2) / A(x) and B(x) * B(x^2) * B(x^4) * ... = 1 / A(x). - Michael Somos, Apr 20 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) - x^(2*k))^(n-k) = Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) + x^(2*k))^(n-k) * (-1)^k = Sum_{n>=0} a(n)*x^(2*n). (End)
From Peter Bala, Jan 05 2021: (Start)
G.f.: Sum_{n = -oo..oo} x^(4*n^2+2*n) * (1 + x^(4*n+1))/(1 - x^(4*n+1)). See Agarwal, p. 285, equation 6.20 with i = j = 1 and mu = 4.
For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/4) = a(n).
If n > 0 and p are coprime then a(n*p + (p^2 - 1)/4) = 0. The proofs are similar to those given for the corresponding results for A115110. Cf. A000729.
For prime p of the form 4*k + 1 and for n not congruent to (p - 1)/4 (mod p) we have a(n*p^2 + (p^2 - 1)/4) = 3*a(n) (since b(n), where b(4*n+1) = a(n), is multiplicative). (End)
From Peter Bala, Mar 22 2021: (Start)
G.f. A(q) satisfies:
A(q^2) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+2)) (set z = q, alpha = q^2, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)) (set z = q^2, alpha = q, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^n/(1 + q^(2*n+1))^2 = Sum_{n = -oo..oo} q^(3*n+1)/(1 + q^(2*n+1))^2. (End)
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) + 2 * Sum_{n>=1, k>=0} (-1)^k * q^(k*(k+2*n+1)+n). - Mamuka Jibladze, May 17 2021
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) * (1 + q^(2*k+1))/(1 - q^(2*k+1)). - Mamuka Jibladze, Jun 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022

More terms and information from Michael Somos, Mar 23 2003

A258210 Expansion of f(-q) * f(-q^2) * chi(-q^3) in powers of q where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, -2, 0, 1, 4, 0, 0, -2, -4, 2, 0, 0, -2, 0, 0, 1, 4, 4, 0, -4, 0, 0, 0, 0, -3, -4, 0, 0, 4, 0, 0, -2, 0, 2, 0, 4, -2, 0, 0, 2, 4, 0, 0, 0, -8, 0, 0, 0, -1, -6, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 1, 8, 0, 0, -4, 0, 0, 0, 4, -2, -4, 0, 0, 0, 0, 0, -4

Offset: 0

Michael Somos, May 23 2015

sign

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

Denoted by a_6(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - q - 2*q^2 + q^4 + 4*q^5 - 2*q^8 - 4*q^9 + 2*q^10 - 2*q^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(d). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.4.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A077285, A104794, A121444, A258228, A258277, A258278, A290217.

For the square of this series see A252650.

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^6] QPochhammer[ q^5, q^6]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ (1/2) EllipticThetaPrime[ 1, 0, q^(1/2)] / EllipticTheta[ 1, Pi/6, q^(1/2)], {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) / eta(x^6 + A), n))};

{a(n) = if( n<1, n==0, (-1)^n * (1 - (n%3==2)*3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))}; /* Michael Somos, Jun 04 2015 */

Expansion of f(-q)^2 * f(-q^6) / f(-q, -q^5) in powers of q where f(,) is Ramanujan's general theta function.
Expansion of eta(q) * eta(q^2) * eta(q^3) / eta(q^6) in powers of q.
Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121444.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^2*k) / (1 + x^(3*k)).
a(n) = (-1)^n * A258228(n). Convolution inverse of A077285.
a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(6*n + 4) = A122856(n). a(12*n + 1) = -1 * A002175(n).
a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A104794(n). a(3*n + 1) = -A258277(n). a(3*n + 2) = -2*A258278(n). - Michael Somos, May 01 2016
G.f.: Product_{i>0} 1/(1 + Sum_{j>0} j*x^(j*i)). - Seiichi Manyama, Oct 08 2017

A113652 Expansion of (1 - theta_4(q)^2) / 4 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 03 2005

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

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(v).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28, Article 269.

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

Cf. A002654, A053692, A008441, A104794, A113407, A122856, A122865, A258277, A258278.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[4, 0, q]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[(1 - EllipticK[m] / (Pi/2)) / 4, {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)

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

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

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==2, 1 - X/(1 - X), 1 / ((1 - X) * (1 - kronecker( -4, p)*X))) )[n])};

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

a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 4), (1 + (-1)^e)/2 if p == 3 (mod 4).
Expansion of (1 - eta(q)^4 / eta(q^2)^2) / 4 in powers of q.
Moebius transform is period 8 sequence [ 1, -2, -1, 0, 1, 2, -1, 0, ...].
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 - 2*u3 + u6 - u1^2 + 3*u3^2 + 2*u1*u3 - 4*u2*u6.
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 + x^k).
G.f.: Sum_{k>0} -(-1)^k * x^k / (1 + x^(2*k)).
G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 + x^(2*k - 1)).
a(n) = -(-1)^n * A002654(n). a(n) = - A104794(n) / 4 unless n = 0.
a(2*n) = - A002654(n). a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 0. a(6*n + 2) = - A122856(n). a(6*n + 4 ) = - A122856(n). - Michael Somos, Jun 06 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(9*n + 3) = a(9*n + 6) = 0. - Michael Somos, Jun 06 2015

A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 15 2012

sign

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

			G.f. = 1 + 2*q - 4*q^5 - 4*q^8 + 2*q^9 - 4*q^13 + 4*q^16 + 4*q^17 + ...

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. A053692, A104794, A113407, A134343, A246950.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)), n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==5) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, e+1, !(e%2))))};

Expansion of eta(q^2)^5 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, -3, 2, -3, 2, -2, ...].
G.f.: Product_{k>0} (1 - x^(2*k))^5 / ((1 - x^k)^2 * (1 - x^(8*k))).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A104794(n). a(4*n + 1) = 2 * A134343(n).
a(n) = (-1)^n * A246950(n). a(8*n + 1) = 2 * A113407(n). a(8*n + 5) = -4 * A053692(n). - Michael Somos, Jun 10 2015

A246950 Expansion of phi(-q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 08 2014

sign

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

			G.f. = 1 - 2*q + 4*q^5 - 4*q^8 - 2*q^9 + 4*q^13 + 4*q^16 - 4*q^17 + ...

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. A053692, A104794, A113407, A134343, A204531.

A := Basis( ModularForms( Gamma1(64), 1), 85); A[1] - 2*A[2] + 4*A[6] - 4*A[9] - 2*A[10] + 4*A[14] + 4*A[17] - 4*A[18] - 6*A[26] + 4*A[30] - 4*A[35] + 4*A[36]; /* Michael Somos, Jun 21 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}];

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

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==1) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, (e+1), !(e%2))))};

Expansion of f(-q, -q) * f(q, -q) in powers of q where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 21 2015
Expansion of eta(q)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -3, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053692.
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k)).
a(n) = (-1)^n * A204531(n). a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
a(8*n) = A104794(n). a(4*n + 1) = - 2 * A134343(n).
a(8*n + 1) = -2 * A113407(n). a(8*n + 5) = 4 * A053692(n). - Michael Somos, Jun 10 2015

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