A089801 -id:A089801 - cp's OEIS frontend

A143064 Expansion of a Ramanujan false theta series variation of A089801 in powers of x.

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, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 21 2008

sign

			G.f. = 1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 + ...
G.f. = q + q^4 - q^16 - q^25 + q^49 + q^64 - q^100 - q^121 + q^169 + q^196 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 89, Equation (1.2), page 100, Equation (5.3)
G. E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See page 3, Equation (1.5)

Cf. A089801, A010054, A143062, A143065.

Column m=0 of A185646.

a[ n_] := With[ {m = Sqrt[3 n + 1]}, If[ IntegerQ @ m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, n}], {x, 0, n}]; (* Michael Somos, Nov 04 2013 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ x, x^2, k], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k / QPochhammer[ -x, x^2, k + 1], {k, 0, 2 n}], {x, 0, 2 n}]; (* Michael Somos, Jun 30 2015 *)

{a(n) = my(m); if( issquare( 3*n + 1, &m), (-1)^(m \ 3) )};

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( e%2, 0, p==2, -(-1)^(e/2), p == 3, 0, p%6 == 1, 1, (-1)^(e/2))))}; /* Michael Somos, Jul 19 2013 */

/* Continued Fraction: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 18 2013

Expansion of Sum_{k>=0} x^k / (Product_{j=0..k} ( 1 + x^(2*k + 1) ) ) in powers of x^2. - Michael Somos, Nov 04 2013
a(n) = b(3*n + 1) where b() is multiplicative with b(p^(2*e)) = -(-1)^e if p = 2, b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = 1 if p = 1 (mod 6), and b(p^(2*e-1)) = b(3^e) = 0 if e>0. - Michael Somos, Jul 19 2013
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
a(8*n) = A143062(n). Convolution of A010054 with A143065. - Michael Somos, Jul 19 2013
G.f.: Sum_{k>=0} (-1)^k * x^(3*k^2 + 2*k) * ( 1 + x^(2*k + 1) ).
G.f.: 1/(1 - x*(1-x)/(1 - x^2*(1-x^2)/(1 - x^3*(1-x^3)/(1 - x^4*(1-x^4)/(1 - ...))))), a continued fraction. - Paul D. Hanna, Jul 18 2013
abs(a(n)) = A089801(n). - Michael Somos, Jun 30 2015
G.f.: 1 + x*(1-x) + x^2*(1-x)*(1-x^3) + x^3*(1-x)*(1-x^3)*(1-x^5) + ... . - Michael Somos, Aug 03 2017

A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

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, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (the present sequence), psi(q) (A010054), chi(q) (A000700).
Theta series of the one-dimensional lattice Z.
Also, essentially the same as the theta series of the one-dimensional lattices A_1, A*_1, D_1, D*_1.
Number of ways of writing n as a square.
Closely related: theta_4(x) = Sum_{m = -oo..oo} (-x)^(m^2). See A002448.
Number 6 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 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + ...

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, Exercise 1, p. 91.
Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 64.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5n].
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.1); p. 78, Eq. (32.22).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth Edition, Clarendon Press, Oxford, 2009, Theorem 352, p. 372.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

T. D. Noe, Table of n, a(n) for n = 0..10000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
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.
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

1st column of A286815. - Seiichi Manyama, May 27 2017
Row d=1 of A122141.
Cf. A002448 (theta_4). Partial sums give A001650.
Cf. A010052, A010054, A089801.
Cf. A000007, A004015, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_3, A_2, A_4, ...).

using Nemo
function JacobiTheta3(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A000122List(len) = JacobiTheta3(len, 1)
A000122List(105) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma0(4), 1/2), 100) [1]; /* Michael Somos, Jun 10 2014 */

L := Lattice("A",1); A := ThetaSeries(L, 20); A; /* Michael Somos, Nov 13 2014 */

add(x^(m^2),m=-10..10): seq(coeff(%,x,n), n=0..100);
# alternative
A000122 := proc(n)
    if n = 0 then
        1;
    elif issqr(n) then
        2;
    else
        0 ;
    end if;
end proc:
seq(A000122(n),n=0..100) ; # R. J. Mathar, Feb 22 2021

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
CoefficientList[ Sum[ x^(m^2), {m, -(n=10), n} ], x ]
SquaresR[1, Range[0, 104]] (* Robert G. Wilson v, Jul 16 2014 *)
QP = QPochhammer; s = QP[q^2]^5/(QP[q]*QP[q^4])^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)
(4 QPochhammer[q^2]/QPochhammer[-1,-q]^2 + O[q]^101)[[3]] (* Vladimir Reshetnikov, Sep 16 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = issquare(n) * 2 -(n==0)}; /* Michael Somos, Jun 17 1999 */

from sympy.ntheory.primetest import is_square
def A000122(n): return is_square(n)<<1 if n else 1 # Chai Wah Wu, May 17 2023

Q = DiagonalQuadraticForm(ZZ, [1])
Q.representation_number_list(105) # Peter Luschny, Jun 20 2014

Expansion of eta(q^2)^5 / (eta(q)*eta(q^4))^2 in powers of q.
Euler transform of period 4 sequence [2, -3, 2, -1, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - v^2 + 2 * w * (w - u). - Michael Somos, Jul 20 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = w^4 - v^4 + w * (u - w)^3. - Michael Somos, May 11 2019
G.f.: Sum_{m=-oo..oo} x^(m^2);
a(0) = 1; for n > 0, a(n) = 0 unless n is a square when a(n) = 2.
G.f.: Product_{k>0} (1 - x^(2*k))*(1 + x^(2*k-1))^2.
G.f.: s(2)^5/(s(1)^2*s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
The Jacobi triple product identity states that for |x| < 1, z != 0, Product_{n>0} {(1-x^(2n))(1+x^(2n-1)z)(1+x^(2n-1)/z)} = Sum_{n=-inf..inf} x^(n^2)*z^n. Set z=1 to get theta_3(x).
For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1))). - Mikael Aaltonen, Jan 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1/(4 t)) = 2^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016
a(n) = A000132(n)(mod 4). - John M. Campbell, Jul 07 2016
a(n) = (2/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = 2 * A010052(n) if n>0. a(3*n + 1) = 2 * A089801(n). a(3*n + 2) = 0. a(4*n) = a(n). a(4*n + 2) = a(4*n + 3) = 0. a(8*n + 1) = 2 * A010054(n). - Michael Somos, May 11 2019
Dirichlet g.f.: 2*zeta(2s). - Francois Oger, Oct 26 2019 [Corrected by Sean A. Irvine, Nov 26 2024]
G.f. appears to equal exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021
From Peter Bala, Sep 27 2023: (Start)
G.f. A(x) satisfies A(x)*A(-x) = A(-x^2)^2.
A(x) = Sum_{n >= 1} x^(n-1)*Product_{k >= n} 1 - (-x)^k.
A(x)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*x^(2*n-1)/(1 - x^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
A(x) = 1 + 2*Sum_{n >= 1} x^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + x^k ) /( Product_{k = 1..n} 1 + x^(2*k) ). See Fine, equation 14.43. (End)

A001082 Generalized octagonal numbers: k(3k-2), k=0, +- 1, +- 2, +-3, ...

Original entry on oeis.org

0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96, 120, 133, 161, 176, 208, 225, 261, 280, 320, 341, 385, 408, 456, 481, 533, 560, 616, 645, 705, 736, 800, 833, 901, 936, 1008, 1045, 1121, 1160, 1240, 1281, 1365, 1408, 1496, 1541, 1633, 1680, 1776, 1825, 1925, 1976

Offset: 1

N. J. A. Sloane and Tom Duff

Numbers of the form 3*m^2+2*m, m an integer.
3*a(n) + 1 is a perfect square.
a(n) mod 10 belongs to a periodic sequence: 0, 1, 5, 8, 6, 1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. - Mohamed Bouhamida, Sep 04 2009
A089801 is the characteristic function. - R. J. Mathar, Oct 07 2011
Exponents of powers of q in one form of the quintuple product identity. (-x^-2 + 1) * q^0 + (x^-3 - x) * q^1 + (-x^-5 + x^3) * q^5 + (x^-6 - x^4) * q^8 + ... = Sum_{n>=0} q^(3*n^2 + 2*n) * (x^(3*n) - x^(-3*n - 2)) = Product_{k>0} (1 - x * q^(2*k - 1)) * (1 - x^-1 * q^(2*k - 1)) * (1 - q^(2*k)) * (1 - x^2 * q^(4*k)) * (1 - x^-2 * q^(4*k - 4)). - Michael Somos, Dec 21 2011
The offset 0 would also be valid here, all other entries of generalized k-gonal numbers have offset 0 (see cross references). - Omar E. Pol, Jan 12 2013
Also, x values of the Diophantine equation x(x+3)+(x+1)(x+2) = (x+y)^2+(x-y)^2. - Bruno Berselli, Mar 29 2013
Numbers n such that Sum_{i=1..n} 2*i*(n-i)/n is an integer (the addend is the harmonic mean of i and n-i). - Wesley Ivan Hurt, Sep 14 2014
Equivalently, integers of the form m*(m+2)/3 (nonnegative values of m are listed in A032766). - Bruno Berselli, Jul 18 2016
Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020
Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020
Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021
From Peter Bala, Nov 26 2024: (Start)
Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).
Exponents of q^2 in the expansion of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

			For the ninth comment: 65 is in the sequence because 65 = 13*(13+2)/3 or also 65 = -15*(-15+2)/3. - _Bruno Berselli_, Jul 18 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
George E. Andrews, Euler's Pentagonal Number Theorem, Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 279-284.
Sandy H. L. Chen, Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in q-Series, Open Journal of Discrete Mathematics, Vol.4, No.2, April 2014.
John Elias, Illustration of initial terms: Hourglass Octagonals.
John Elias, Illustration: Pentagonal-Octagonal Star Configurations
Ralf Stephan, On the solutions to 'px+1 is square'.
Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv preprint arXiv:1503.03743 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Partial sums of A022998.
Cf. A000567, A005563, A085785, A089801, A245031.
Column 4 of A195152. A045944.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), this sequence (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

a001082 n = a001082_list !! n
a001082_list = scanl (+) 0 $ tail a022998_list
-- Reinhard Zumkeller, Mar 31 2012

[n^2 - n - Floor(n/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Sep 14 2014

seq(n*(n-1)-floor(n/2)^2, n=1..51); # Gary Detlefs, Feb 23 2010

Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}]
LinearRecurrence[{1,2,-2,-1,1},{0,1,5,8,16},60] (* Harvey P. Dale, Feb 03 2024 *)

{a(n) = if( n%2, (n-1) * (3*n + 1) / 4, n * (3*n - 4) / 4)};

a(n) = n*(3*n-4)/4 if n even, (n-1)*(3*n+1)/4 if n odd.
a(n) = n^2 - n - floor(n/2)^2.
G.f.: Sum_{n>=0} (-1)^n*[x^(a(2n+1)) + x^(a(2n+2))] = 1/1 - (x-x^2)/1 - (x^2-x^4)/1 - (x^3-x^6)/1 - ... - (x^k - x^(2k))/1 - ... (continued fraction where k=1..inf). - Paul D. Hanna, Aug 16 2002
a(n+1) = ceiling(n/2)^2 + A046092(floor(n/2)).
a(2n) = n(3n-2) = A000567(n), a(2n+1) = n(3n+2) = A045944(n). - Mohamed Bouhamida, Nov 06 2007
O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar, Apr 15 2008
a(n) = n^2+n-ceiling(n/2)^2 with offset 0 and a(0)=0. - Gary Detlefs, Feb 23 2010
a(n) = (6*n^2-6*n-1-(2*n-1)*(-1)^n)/8. - Luce ETIENNE, Dec 11 2014
E.g.f.: (3*x^2*exp(x) + x*exp(-x) - sinh(x))/4. - Ilya Gutkovskiy, Jul 15 2016
Sum_{n>=2} 1/a(n) = (9 + 2*sqrt(3)*Pi)/12. - Vaclav Kotesovec, Oct 05 2016
Sum_{n>=2} (-1)^n/a(n) = 3*log(3)/2 - 3/4. - Amiram Eldar, Feb 28 2022

New sequence name from Matthew Vandermast, Apr 10 2003
Editorial changes by N. J. A. Sloane, Feb 03 2012
Edited by Omar E. Pol, Jun 09 2012

A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 14 2006

nonn

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

			G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + 2*x^13 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^17 + 2*q^20 + 2*q^26 + 2*q^29 + q^32 + ...

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. A002654, A035154, A089801, A113446, A121444, A122865, A258278.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 2}, Sum[ KroneckerSymbol[ -4, d], {d, Divisors@m}]]]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = (QP[q^2]^2*QP[q^3]*(QP[q^12]/(QP[q]*QP[q^4]*QP[q^6]) ))^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3])^2 / (4 x^(2/3)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)])^2 / (2 x^(3/4)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^2, n))};

Expansion of (chi(x) * psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^2 in powers of q.
Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, -2, ...].
a(4*n + 3) = a(8*n + 4) = 0. a(n) = A002654(3*n + 2) = A035154(3*n + 2) = A113446(3*n + 2). a(2*n) = A122865(n). a(4*n + 1) = 2 * A121444(n). a(4*n + 2) = A122856(n).
a(n) = (-1)^n * A258278(n). Convolution square of A089801.

A089802 Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.

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

Offset: 0

Eric W. Weisstein, Nov 12 2003

sign

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

Number 10 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..10000
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.

Cf. A001082, A002448, A089801, A089812.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := (-1)^n Sign @ SquaresR[ 1, 3 n + 1]; (* Michael Somos, Jun 30 2015 *)

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

Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(-x) * psi(x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Dec 23 2011
Expansion of f(-x, -x^5) in powers of x, where f(, ) is Ramanujan's general theta function.
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089812. - Michael Somos, Dec 23 2011
Euler transform of period 6 sequence [-1, 0, 0, 0, -1, -1, ...]. - Michael Somos, Apr 12 2005
abs(a(n)) is the characteristic function of A001082. - Michael Somos, Oct 31 2005
G.f.: Sum_{k in Z} (-1)^k * x^((3*k^2 - 2*k)) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - Michael Somos, Oct 31 2005
A002448(3*n + 1) = -2 * a(n). - Michael Somos, Jul 07 2006
a(n) = (-1)^n * A089801(n).
a(n) = -(1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Corrected by N. J. A. Sloane, Nov 05 2005

A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos and Omar E. Pol, Jul 09 2012

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

Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.

			G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...

Seiichi Manyama, 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.
Index entries for characteristic functions.

Cf. A047621, A074377, A115359.
A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.
Cf. A000122, A000700, A121373.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

PARI
```
{a(n) = issquare(16*n + 9)};
```

Euler transform of period 16 sequence [ 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, -1, ...].
G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).
a(n) = A010054(2*n + 1) = A115359(2*n).
Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024

A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 11 2007

nonn

			G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...
G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions.

Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

Cf. A085787, A113429.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* Michael Somos, Jan 30 2017 *)
a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* Michael Somos, Jan 30 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1,n, 1 + x^k*[-1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

PARI
```
{a(n) = issquare( 40*n + 9)};
```

f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
The characteristic function of A085787 generalized heptagonal numbers.
Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].
G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)) = Sum_{k in Z} x^((5*k^2 + 3*k) / 2).
a(n) = |A113429(n)|. a(3*n + 2) = 0.
Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - Amiram Eldar, Jan 13 2024

A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 1, 2, 4, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 8, 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, 4, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 0

Offset: 0

Michael Somos, May 24 2015

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

			G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 + 4*q^9 + 2*q^10 + 2*q^13 + q^16 + ...

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. A002175, A019670, A089801, A121444, A122856, A122856, A256282, A258260.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + A[2] + A[3] + A[5] + 2*A[6] + A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] + 2*A[18] + 4*A[19];

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[ #, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + n + #) &]];
a[ n_] := If[ n < 2, Boole[n >= 0], Times @@ (Which[ # == 2, 1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)]), {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{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%4==1, e+1, if( p==3, 4, 1) * (1 - e%2) )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18) / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, ...].
Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 4, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(2*n) = a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 3) = 0. a(12*n + 1) = A002175. a(12*n + 5) = 2 * A121444(n).
a(n) = Sum_{d|n} A258260(d) * (-1)^(n+d) if n>0.
a(n) = (-1)^n * A256282(n). - Michael Somos, Jun 06 2015
a(n) is multiplicative with a(0) = 1, a(2^e) = 1, a(3^e) = 2*(1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015
Expansion of A0(x)^2 + A0(x)*A1(x) + A1(x)^2 in powers of x where A0(x) = phi(x^9), A1(x) = x * f(x^3, x^15) = x * A089801(x^3). - Michael Somos, Jun 23 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Nov 24 2023

A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, 0, -2, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^5, b = -x. - Michael Somos, Jul 12 2012
Convolution square is A258279. - Michael Somos, May 25 2015
Number 8 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 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan 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.

Cf. A002448, A080995, A089801, A089802, A089807, A089812, A204843, A204853, A258279.

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q], {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q])  /2, {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = QP[q^2]^2*(QP[q^3] / (QP[q]*QP[q^6])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = my(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))}; /* Michael Somos, Nov 05 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 26 2008 */

Expansion of Jacobi theta function (3theta_4(q^9) - theta_4(q)) / 2 in powers of q.
a(n) is multiplicative with  a(0)=1, a(2^e) = -(1 + (-1)^e)/2, if e>0, a(3^e) = -2(1 + (-1)^e)/2 if e>0, a(p^e) = (1 + (-1)^e)/2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, -1, ...].
G.f.: (Sum_{k in Z} 3 * (-x)^((3*k)^2) - (-x)^(k^2)) / 2 = Product_{k>0} (1 - x^(2*k)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k-5))).
Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. (End)
Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2007
Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function.
From Michael Somos, Sep 17 2007: (Start)
Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.
Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two-variable theta function. (End)
From Michael Somos, Jan 26 2008: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A080995.
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). (End)
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(8*n + 5) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = A089802(n). a(4*n) = A089807(n). a(9*n) = A002448(n).
a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(abs(2-4*sin((floor(sqrt(n))+1)*Pi/3)^2) - 4*sin((floor(sqrt(n))+2)*Pi/3)^2)*(-1)^floor(floor(sqrt(n)-1)/3). - Mikael Aaltonen, Jan 17 2015
From Michael Somos, May 25 2015: (Start)
a(n) = (-1)^n * A089807(n) = A204843(4*n) = A204853(4*n).
a(8*n + 1) = A089812(n). a(12*n + 4) = - A089801(n). (End)
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 2, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 0

Eric W. Weisstein, Nov 12 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^5, b = x. - Michael Somos, Jul 12 2012
Number 11 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 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan 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.

Related to the 14 primitive eta-products which are holomorphic modular forms of weight 1/2: A000122, A002448, A010054, A010815, A080995, A089801, A089802, this sequence, A089810, A089812, A106459, A121373, A133985, A133988. - Seiichi Manyama, May 15 2017

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, 0, Sqrt[ -q]] / (2 (-q)^(1/8)), {q, 0, n}] (* Michael Somos, Jul 12 2012 *);

{a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */

a(n) = -b(n) where b() is multiplicative with b(3^e) = -2(1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...].
G.f.: (Sum_{k in Z} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))). (End)
Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26 2006
Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007
Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089801.
a(n) = (-1)^n * A089810(n). - Michael Somos, Jan 20 2012
For n > 0, a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(2-4*sin(floor(sqrt(n))*Pi/3)^2). - Mikael Aaltonen, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

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A143064 Expansion of a Ramanujan false theta series variation of A089801 in powers of x.

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A000122 Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

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A001082 Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...

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A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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A089802 Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.

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A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

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A001082 Generalized octagonal numbers: k(3k-2), k=0, +- 1, +- 2, +-3, ...