A030203 -id:A030203 - cp's OEIS frontend

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168

Offset: 1

N. J. A. Sloane

Coefficients of the cusp form of weight 12 for the full modular group.
It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 (mod p) are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007
Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016
For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017
Conjecture: |a(n)| with n > 1 can never be a perfect power. This has been verified for n up to 10^6. - Zhi-Wei Sun, Dec 18 2024
Conjecture: The numbers |a(n)| (n = 1,2,3,...) are distinct. This has been verified for the first 10^6 terms. - Zhi-Wei Sun, Dec 21 2024
Conjecture: |a(n)| > 2*n^4 for all n > 2. This has been verified for n = 3..10^6. - Zhi-Wei Sun, Dec 25 2024
Conjecture: a(m)^2 + a(n)^2 can never be a perfect power. This implies Lehmer's conjecture that a(n) is never zero. We have verified that there is no perfect power among a(m)^2 + a(n)^2 with m,n <= 1000 . - Zhi-Wei Sun, Dec 28 2024
Conjecture: The equation |a(m)a(n)| = x^k with m < n, k > 1 and x >= 0 has no solution. This has been verified for m < n <= 5000. - Zhi-Wei Sun, Dec 29 2024
For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - Zhi-Wei Sun, Jan 25 2025

			G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...
35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - _Wolfdieter Lang_, Jan 21 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.
Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
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).
H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.

Simon Plouffe, Table of n, a(n) for n = 1..16090
Jennifer S. Balakrishnan, William Craig, and Ken Ono, Variations of Lehmer's Conjecture for Ramanujan's tau-function, arXiv:2005.10345 [math.NT], 2020.
Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, Even values of Ramanujan's tau-function, arXiv:2102.00111 [math.NT], 2021.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
François Brunault, La fonction Tau de Ramanujan. [Wayback Machine link]
Denis Xavier Charles, Computing The Ramanujan Tau Function.
Benoit Cloitre, On the fractal behavior of primes, 2011.
John Cremona, Home page.
Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], 2013-2014.
Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
John A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
John A. Ewell, Ramanujan's Tau Function.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Luis H. Gallardo, On some formulae for Ramanujan's tau function, Rev. Colomb. Matem. 44 (2010) 103-112.
M. Z. Garaev, V. C. Garcia, and S. V. Konyagin, Waring problem with the Ramanujan tau function, Izvestiya: Mathematics, Vol. 72, No 1 (2008), pp. 35-46; arXiv preprint, arXiv:math/0607169 [math.NT], 2006.
Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan's tau function, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; arXiv preprint, arXiv:1606.08037 [math.CO], 2016.
Hansraj Gupta, The Vanishing of Ramanujan's Function(n), Current Science, 17 (1948), p. 180. [Wayback Machine link]
James Lee Hafner and Jeffrey Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
Yang-Hui He and John McKay, Moonshine and the Meaning of Life, in: M. Bhargava et al., Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694, American Mathematical Society, 2017; arXiv preprint, arXiv:1408.2083 [math.NT], 2014.
Michael J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317; arXiv preprint, arXiv:math/0212397 [math.AT], 2002.
Masanobu Kaneko and Don Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
Jon Keating and Brady Haran, The Key to the Riemann Hypothesis, Numberphile video (2016).
Jerry B. Keiper, Ramanujan's Tau-Dirichlet Series [Dead link?]
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
N. Laptyeva and V. K. Murty, Fourier coefficients of forms of CM-type, Indian Journal of Pure and Applied Mathematics, Volume 45, Issue 5 (October 2014), pp. 747-758.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.
LMFDB, Newform orbit 1.12.a.a
Florian Luca and Igor E. Shparlinski, Arithmetic properties of the Ramanujan function, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 116. No. 1 (2006), pp. 1-8; arXiv preprint, arXiv:math/0607591 [math.NT], 2006.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
Yuri Matiyasevich, Computational rediscovery of Ramanujan's tau numbers, Integers (2018) 18A, Article #A14.
Keith Matthews, Computing Ramanujan's tau function.
Stephen C. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.
Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
Pieter Moree, On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions, arXiv:math/0201265 [math.NT], 2002.
M. Ram Murty and V. Kumar Murty, The Ramanujan tau-function, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23.
M. Ram Murty, V. Kumar Murty, and T. N. Shorey, Odd values of the Ramanujan tau-function, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
Douglas Niebur, A formula for Ramanujan's tau-function, Illinois Journal of Mathematics, vol.19, no.3, pp.448-449, (1975). - _Joerg Arndt_, Sep 06 2015
Oklahoma State Mathematics Department, Ramanujan tau L-Function. [broken link]
Jon Perry, Ramanujan's Tau Function. [Wayback Machine link]
Simon Plouffe, The first 225035 terms (432 MB) [Wayback Machine link]
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
Srinivasa Ramanujan, Collected Papers, Table of tau(n);n=1 to 30.
Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's tau function, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's Tau function, 1997.
Jean-Pierre Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, My Favorite Integer Sequences, arXiv:math/0207175 [math.CO], 2002.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
David A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n), 1998.
William A. Stein, Database.
Zhi-Wei Sun, Questions on the reciprocals of the values of the tau function, Question 484763 at MathOverflow, December 25, 2024.
Zhi-Wei Sun, Pythagorean triples and Ramanujan's tau function, Question 484936 at MathOverflow, December 28, 2024.
Zhi-Wei Sun, Additive combinatorics for Ramanujan's tau function, Question 485138 at MathOverflow, January 1, 2025.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
F. Van der Blij, The function tau (n) of S. Ramanujan (an expository lecture), Math. Student, Vol. 18, No. 3 (1950), pp. 83-99; entire issues 3 and 4.
Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
G. N. Watson, A table of Ramanujan's function tau(n), Proc. London Math. Soc., 51 (1950), 1-13.
Eric Weisstein's World of Mathematics, Tau Function.
Kenneth S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35; author's copy.
Index entries for "core" sequences.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m.
Index entries for sequences related to Chebyshev polynomials.

Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
For a(n) mod N for various values of N see A046694, A098108, A126812-...
Cf. A006352, A099059, A099060, A262339, A292781.
For primes p such that tau(p) == -1 (mod 23) see A106867.
Cf. A010816, A004009. A013973.
Cf. A126832(n) = a(n) mod 5.

using Nemo
function DedekindEta(len, r)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(r, len, z)
    [coeff(e, j) for j in 0:len - 1] end
RamanujanTauList(len) = DedekindEta(len, 24)
RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018

M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);

Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */

M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);

CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)
max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)
RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};

{a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2,(-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};

taup(p,e)={
    if(e==1,
        (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
    ,
        my(t=taup(p,1));
        sum(j=0,e\2,
            (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
        )
    )
};
a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
\\ Charles R Greathouse IV, Apr 22 2013

\\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));
vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015

a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end
p A000594(100) # Seiichi Manyama, Mar 26 2017

def A000594(n)
  ary = [0, 1]
  (2..n).each{|i|
    s, t, u = 0, 1, 0
    (1..n).each{|j|
      t += 9 * j
      u += j
      break if i <= u
      s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
    }
    ary << s / (i - 1)
  }
  ary[1..-1]
end
p A000594(100) # Seiichi Manyama, Nov 25 2017

CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013

list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016

G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011
abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.
Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004
G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013
a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015
a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016
Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Product_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s  + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017
a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n)). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. - Wolfdieter Lang, Jan 27 2017
G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017
a(n) (mod 5) == A126832(n).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
a(n) = n^4*sigma(n)-24*Sum_{k=1..n-1} (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025

A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

Original entry on oeis.org

1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0, 0, 4, 0, 8, 0, 4, 8, 0, 0, 8, 8, 0, 0, 0, 8, 0, 0, 0, 4, 12, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 4, 16, 0, 0, 8, 0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 8, 4, 8, 0, 0, 16, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 8, 4, 0, 12, 8

Offset: 0

N. J. A. Sloane

Number of points in square lattice on the circle of radius sqrt(n). Equivalently, number of Gaussian integers of norm n (cf. Conway-Sloane, p. 106).
Let b(n)=A004403(n), then Sum_{k=1..n} a(k)*b(n-k) = 1. - John W. Layman
Theta series of D_2 lattice.
Number 6 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The zeros in this sequence correspond to those integers with an equal number of 4k+1 and 4k+3 divisors, or equivalently to those that have at least one 4k+3 prime factor with an odd exponent (A022544). - Ant King, Mar 12 2013
If A(q) = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + ... denotes the o.g.f. of this sequence then the function F(q) := 1/4*(A(q^2) - A(q^4)) = ( Sum_{n >= 0} q^(2*n+1)^2 )^2 is the o.g.f. for counting the ways a positive integer n can be written as the sum of two positive odd squares. - Peter Bala, Dec 13 2013
Expansion coefficients of (2/Pi)*K, with the real quarter period K of elliptic functions, as series of the Jacobi nome q, due to (2/Pi)*K = theta_3(0,q)^2. See, e.g., Whittaker-Watson, p. 486. - Wolfdieter Lang, Jul 15 2016
Sum_{k=0..n} a(n) = A057655(n). Robert G. Wilson v, Dec 22 2016
Limit_{n->oo} (a(n)/n - Pi*log(n)) = A062089: Sierpinski's constant. - Robert G. Wilson v, Dec 22 2016
The mean value of a(n) is Pi, see A057655 for more details. - M. F. Hasler, Mar 20 2017

			G.f. = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + 8*q^10 + 8*q^13 + 4*q^16 + 8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + ... . - _John Cannon_, Dec 30 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, p. 32, Lemma 2 (with the proof), p. 116, (9.10) first formula.
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. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
W. König and J. Sprekels, Karl Weierstraß (1815-1897), Springer Spektrum, Wiesbaden, 2016, p. 186-187 and p. 280-281.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 244-245.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
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.
S. Cooper and M. Hirschhorn, A combinatorial proof of a result from number theory, Integers 4 (2004), #A09.
Michael Gilleland, Some Self-Similar Integer Sequences
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62. (This sequence is called rho, see page 6)
M. D. Hirschhorn, Jacobi's Two-Square Theorem and Related Identities
M. D. Hirschhorn, Arithmetic Consequences of Jacobi's Two-Squares Theorem
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Jacobi-Legendre letters, Correspondance mathématique entre Legendre et Jacobi, J. Reine Angew. Math. 80 (1875) 205-279, letter of September 9, 1828, p. 240-243, formula for 2K/Pi on p. 242.
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. [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.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
G. Nebe and N. J. A. Sloane, Home page for this lattice
F. Richman, Counting Gaussian integers in a disk
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
G. Villemin, Sommes de 2 carrés
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Theta Series
Zach Wissner-Gross a.k.a. The Riddler, Riddler Express: Can You Climb Your Way To Victory?, FiveThirtyEight, Sep 24 2021.
G. Xiao, Two squares
Index entries for sequences related to sums of squares
Index entries for sequences mentioned by Glaisher
Index entries for "core" sequences

Row d=2 of A122141 and of A319574, 2nd column of A286815.
Partial sums - 1 give A014198.
A071385 gives records; A071383 gives where records occur.
Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962, A002654,
A104271, A105673.

# JacobiTheta3 is defined in A000122.
A004018List(len) = JacobiTheta3(len, 2)
A004018List(102) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 1), 100) [1]; /* Michael Somos, Jun 10 2014 */

(sum(x^(m^2),m=-10..10))^2;
# Alternative:
A004018list := proc(len) series(JacobiTheta3(0, x)^2, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end:
t1 := A004018list(102);
r2 := n -> t1[n+1]; # Peter Luschny, Oct 02 2018

SquaresR[2,Range[0,110]] (* Harvey P. Dale, Oct 10 2011 *)
a[ n_] := SquaresR[ 2, n]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2, {q, 0, n}]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 Sum[ KroneckerSymbol[-4, d], {d, Divisors@n}]]; (* or *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10/(QPochhammer[ q] QPochhammer[ q^4])^4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = polcoeff( 1 + 4 * sum( k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}; /* Michael Somos, Mar 14 2003 */

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Jul 19 2004 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 1], n)[n])}; /* Michael Somos, May 13 2005 */

a(n)=if(n==0,return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, if(f[i,2]%2 && f[i,1]>2, 0, 1))) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import factorint
def a(n):
    if n == 0: return 1
    an = 4
    for pi, ei in factorint(n).items():
       if pi%4 == 1: an *= ei+1
       elif pi%4 == 3 and ei%2: return 0
    return an
print([a(n) for n in range(102)]) # Michael S. Branicky, Sep 24 2021

from math import prod
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1 # Chai Wah Wu, Jul 07 2022, corrected Jun 21 2024.

Q = DiagonalQuadraticForm(ZZ, [1]*2)
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014

Expansion of theta_3(q)^2 = (Sum_{n=-oo..+oo} q^(n^2))^2 = Product_{m>=1} (1-q^(2*m))^2 * (1+q^(2*m-1))^4; convolution square of A000122.
Factor n as n = p1^a1 * p2^a2 * ... * q1^b1 * q2^b2 * ... * 2^c, where the p's are primes == 1 (mod 4) and the q's are primes == 3 (mod 4). Then a(n) = 0 if any b is odd, otherwise a(n) = 4*(1 + a1)*(1 + a2)*...
G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
a(n) = 4*A002654(n), n > 0.
Expansion of eta(q^2)^10 / (eta(q) * eta(q^4))^4 in powers of q. - Michael Somos, Jul 19 2004
Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * 4 * w. - Michael Somos, Jul 19 2004
Euler transform of period 4 sequence [4, -6, 4, -2, ...]. - Michael Somos, Jul 19 2004
Moebius transform is period 4 sequence [4, 0, -4, 0, ...]. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i) f(t) where q = exp(2 Pi i t).
The constant sqrt(Pi)/Gamma(3/4)^2 produces the first 324 terms of the sequence when expanded in base exp(Pi), 450 digits of the constant are necessary. - Simon Plouffe, Mar 03 2011
a(n) = A004531(4*n). a(n) = 2*A105673(n), if n>0.
Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([+1/2, +1/2], [+1], s) (expansion of 2/Pi*ellipticK(k) in powers of q). - Joerg Arndt, Aug 15 2011
Dirichlet g.f. Sum_{n>=1} a(n)/n^s = 4*zeta(s)*L_(-4)(s), where L is the D.g.f. of the (shifted) A056594. [Raman. J. 7 (2003) 95-127]. - R. J. Mathar, Jul 02 2012
a(n) = floor(1/(n+1)) + 4*floor(cos(Pi*sqrt(n))^2) - 4*floor(cos(Pi*sqrt(n/2))^2) + 8*Sum_{i=1..floor(n/2)} floor(cos(Pi*sqrt(i))^2)*floor(cos(Pi*sqrt(n-i))^2). - Wesley Ivan Hurt, Jan 09 2013
From Wolfdieter Lang, Aug 01 2016: (Start)
A Jacobi identity: theta_3(0, q)^2 = 1 + 4*Sum_{r>=0} (-1)^r*q^(2*r+1)/(1 - q^(2*r+1)). See, e.g., the Grosswald reference (p. 15, p. 116, but p. 32, Lemma 2 with the proof, has the typo r >= 1 instead of r >= 0 in the sum, also in the proof). See the link with the Jacobi-Legendre letter.
Identity used by Weierstraß (see the König-Sprekels book, p. 187, eq. (5.12) and p. 281, with references, but there F(x) from (5.11) on p. 186 should start with nu =1 not 0): theta_3(0, q)^2 = 1 + 4*Sum_{n>=1} q^n/(1 + q^(2*n)). Proof: similar to the one of the preceding Jacobi identity. (End)
a(n) = (4/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
G.f.: Theta_3(q)^2 = hypergeometric([1/2, 1/2],[1],lambda(q)), with lambda(q) = Sum_{j>=1} A115977(j)*q^j. See the Kontsevich and Zagier link, with Theta -> Theta_3, z -> 2*z and q -> q^2. - Wolfdieter Lang, May 27 2018

A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Theta series of lattice C2 with Gram matrix [ 1, 0; 0, 2]. a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Number 17 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by |a_4(n)| in Kassel and Reutenauer 2015. - Michael Somos, Jun 16 2015

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 114 Entry 8(iii).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 102, eq. 9.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

John Cannon, Table of n, a(n) for n = 0..10000
George E. Andrews, Richard Lewis, and Zhi-Guo Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
Michael Gilleland, Some Self-Similar Integer Sequences.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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. [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.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002325, A002479, A028572, A033761, A082564, A112603, A113411, A133692, A139093, A226225, A226240, A242609, A245572, A247719.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), this sequence (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(8), 1), 105); A[1] + 2*A[2] + 2*A[3]; /* Michael Somos, Aug 29 2014 */

d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1,8,n)+d(3,8,n)-d(5,8,n)-d(7,8,n)),n=1..120)];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Sep 09 2012 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^4])^3 / (QPochhammer[ q] QPochhammer[ q^8])^2, {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, 2 * (issquare(n) - issquare(2*n) + 2 * sum( i=1, sqrtint(n\2), issquare(n - 2*i^2))))};

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Aug 23 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])}; /* Michael Somos, Aug 23 2005 */

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

Q = DiagonalQuadraticForm(ZZ, [1,2]); Q.representation_number_list(104); # Peter Luschny, Jun 20 2014

Fine gives an explicit formula for a(n) in terms of the divisors of n.
Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].
Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.
Coefficients in expansion of Sum_{i,j=-inf..inf} q^(i^2 + 2*j^2).
G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-2, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
G.f.: theta_3(q) * theta_3(q^2) = Product_{k>0} (1 + x^(2*k)) * ((1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)))^2.
From Michael Somos, Oct 23 2006: (Start)
Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 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) = (u1 - 3*u3) * (u1 - u2 - u3 + u6) - (u2 - 3*u6) * (u1 - 2*u2 - u3 + 2*u6). (End)
a(n) = 2 * A002325(n) unless n = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2012
From Michael Somos, Aug 29 2014: (Start)
Expansion of phi(q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
a(2*n) = a(n). a(2*n + 1) = 2 * A113411(n). (End)
From Michael Somos, May 17 2015: (Start)
a(n) = A028572(4*n) = A133692(2*n) = A139093(8*n) = A226225(8*n) = A226240(4*n) = A242609(4*n) = A245572(4*n) / 3 = (-1)^floor((n + 1)/2) * A082564(n).
a(8*n + 5) = a(8*n + 7) = 0. a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = 4 * A033761(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(2^e) = 1, b(p^e) = e + 1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8). - Jianing Song, Sep 04 2018 [Corrected by Jeremy Lovejoy, Nov 12 2024]
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/sqrt(2) = 2.221441... (A247719). - Amiram Eldar, Dec 16 2023

A000727 Expansion of Product_{k >= 1} (1 - x^k)^4.

Original entry on oeis.org

1, -4, 2, 8, -5, -4, -10, 8, 9, 0, 14, -16, -10, -4, 0, -8, 14, 20, 2, 0, -11, 20, -32, -16, 0, -4, 14, 8, -9, 20, 26, 0, 2, -28, 0, -16, 16, -28, -22, 0, 14, 16, 0, 40, 0, -28, 26, 32, -17, 0, -32, -16, -22, 0, -10, 32, -34, -8, 14, 0, 45, -4, 38, 8, 0, 0, -34, -8, 38, 0, -22, -56, 2, -28, 0, 0, -10, 20, 64, -40, -20, 44

Offset: 0

N. J. A. Sloane

Number 51 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan (see the link, pp. 155 and 157 Nr. 23.) conjectured the expansion coefficients called Psi_4(n) of eta^4(6*z) in powers of q = exp(2*Pi*i*z), Im(z) > 0, where i is the imaginary unit. In the Finch link on p. 5, multiplicity is used and Psi_4(p^r), called f(p^r), is given (see also b(p^e) formula given by Michael Somos, Aug 23 2006). Mordell proved this conjecture on pp. 121-122 based on Klein-Fricke, Theorie der elliptischen Modulfunktionen, 1892, Band II, p. 374. The product formula for the Dirichlet series, Mordell, eq. (7) for a=2,is used to find Psi_4(n), called f_2(n), from f_2(p) for primes p. The primes p = 2 and 3 do not appear in the product. - Wolfdieter Lang, May 03 2016

			G.f. = 1 - 4*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 + 9*x^8 + ...
G.f. = q - 4*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...

Morris Newman, A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.2.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016).
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.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
S. Ramanujan, On certain Arithmetical Functions, Trans. Cambridge Philos. Soc. 22 (1916) 159-184; also in Collected papers, eds. G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, Chelsea publ. Comp., 1962 (reprint from CUP 1927), pp. 136-162, 340- 341.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000731, A002107, A023003, A187076, A187150, A258404, A280328.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

# DedekindEta is defined in A000594.
L000727List(len) = DedekindEta(len, 4)
L000727List(82) |> println # Peter Luschny, Mar 09 2018

qEigenform( EllipticCurve( [0, 0, 0, 0, 1]), 493); /* Michael Somos, Jun 12 2014 */

A := Basis( ModularForms( Gamma0(36), 2), 493); A[2] - 4*A[8]; /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(36), 2), 493)[1]; /* Michael Somos, May 17 2015 */

Coefficients(&*[(1-x^m)^4:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> -4): seq(a(n), n=0..81); # Alois P. Heinz, Sep 08 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[-4&]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4, {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)
nmax = 80; CoefficientList[Series[Sum[Sum[(-1)^(k+m) * (2*k+1) * q^(k*(k+1)/2 + m*(3*m-1)/2), {k, 0, nmax}], {m, -nmax, nmax}], {q, 0, nmax}], q] (* Vaclav Kotesovec, Dec 06 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-1)^(e/2) * p^(e/2)), for( y=1, sqrtint(p\3), if( issquare( p - 3*y^2, &x), break)); a0=1; if( x%3!=1, x=-x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0=a1; a1=y); a1)))}; /* Michael Somos, Aug 23 2006 */

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

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, 1], 1), 6*n + 1))}; /* Michael Somos, Jul 01 2004 */

ModularForms( Gamma0(36), 2, prec=493).0; # Michael Somos, Jun 12 2014

Euler transform of period 1 sequence [-4, -4, ...]. - Michael Somos, Apr 02 2005
Given g.f. A(x), then B(q) = q * A(q^3)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w*u^2 - v^3 + 16 * u*w^2. - Michael Somos, Apr 02 2005
a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2*x where p = x^2 + 3*y^2 == 1 (mod 6) and x == 1 (mod 3). - Michael Somos, Aug 23 2006
Coefficients of L-series for elliptic curve "36a1": y^2 = x^3 + 1. - Michael Somos, Jul 01 2004
a(n) = (-1)^n * A187076(n). a(2*n + 1) = -4 * A187150(n). a(25*n + 9) = a(25*n + 14) = a(25*n + 19) = a(25*n + 24) = 0. a(25*n + 4) = -5 * a(n). Convolution inverse of A023003. Convolution square of A002107. Convolution square is A000731.
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-4*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 5 (mod 6). Then a( M^2*n + (M^2 - 1)/6 ) = (-1)^k*M*a(n). See Cooper et al., equation 4. - Peter Bala, Dec 01 2020
a(n) = b(6*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(e+1) - ((x-sqrt(-3)*y)/2)^(e+1))/x if p == 1 (mod 3) where p = x^2 + 3*y^2 and x == 1 (mod 3). - Jianing Song, Mar 19 2022

A006571 Expansion of qProduct_{k>=1} (1-q^k)^2(1-q^(11*k))^2.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2, 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0, -4, 0, -8, -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14, 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, -4, 1, 16, -6, 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, -8

Offset: 1

N. J. A. Sloane

Number 23 of the 74 eta-quotients listed in Table I of Martin (1996).
Unique cusp form of weight 2 for congruence group Gamma_1(11). - Michael Somos, Aug 11 2011
For some elliptic curves with p-defects given by this sequence, and for more references, see A272196. See also the Michael Somos formula from May 23 2008 below. - Wolfdieter Lang, Apr 25 2016

			G.f.: q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + ...

Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.
M. du Sautoy, Review of "Love and Math: The Heart of Hidden Reality" by Edward Frenkel, Nature, 502 (Oct 03 2013), p. 36.
N. D. Elkies, Elliptic and modular curves..., in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 42.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 412.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Wiles, Modular forms, elliptic curves and Fermat's last theorem, pp. 243-245 of Proc. Intern. Congr. Math. (Zurich), Vol. 1, 1994.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1002 terms from T. D. Noe)
J. Cowles, Some congruence properties of three well-known sequences: Two notes, J. Num. Theory 12(1) (1980) 84.
H. Darmon, A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices Amer. Math. Soc., Dec. 1999, pp. 1397-1401.
F. Diamond, Congruences between modular forms: raising the level and dropping Euler factors, in Elliptic curves and modular forms (Washington, DC, 1996). Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 11143-11146.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
A. W. Knapp, Review of "Love and Math: The Heart of Hidden Reality" by E. Frenkel, Notices Amer. Math. Soc., 61 (2014), pp. 1056-1060; see p. 1058, but beware typos.
LMFDB, Newform orbit 11.2.a.a
LMFDB, Elliptic curve with LMFDB label 11.a3 (Cremona label 11a3)
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Shimura, Goro, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220.
G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A002070 (terms with prime indices), A032442, A030200.

[ Coefficient(qEigenform(EllipticCurve([0, -1, 1, 0, 0]), n+1),n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 29 2007 */

[ Coefficient(Basis(ModularForms(Gamma0(11), 2))[2], n) : n in [1..100] ]; /* Klaus Brockhaus, Jan 31 2007 */

Basis( CuspForms( Gamma1(11), 2), 101) [1]; /* Michael Somos, Jul 14 2014 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)
a[ n_] := SeriesCoefficient[ q (Product[ (1 - q^k), {k, 11, n, 11}] Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, ellak( ellinit( [0, -1, 1, 0, 0], 1), n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, 1, a0=1; a1 = y = -sum(x=0, p-1, kronecker( 4*x^3 - 4*x^2 + 1, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma1(11), 2, prec = 101).0 # Michael Somos, Aug 11 2011

Expansion of (eta(q) * eta(q^11))^2 in powers of q.
a(n) == A000594(n) (mod 11). [Cowles]. - R. J. Mathar, Feb 13 2007
Euler transform of period 11 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...]. - Michael Somos, Feb 12 2006
a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) for p != 11. - Michael Somos, Feb 12 2006
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u*w * (u + 4*v + 4*w) - v^3. - Michael Somos, Mar 21 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 Pi i t).
Convolution square of A030200.
Coefficients of L-series for elliptic curve "11a3": y^2 + y = x^3 - x^2. - Michael Somos, May 23 2008
Convolution inverse is A032442. - Michael Somos, Apr 21 2015
a(prime(n)) = prime(n) - A272196(n), n >= 3.
  a(2) = -2 is not  2 - A272196(1) = 0. Modularity pattern of some elliptic curves. - Wolfdieter Lang, Apr 25 2016

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

A000731 Expansion of Product (1 - x^k)^8 in powers of x.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 0, -125, -160, 308, 0, 110, 0, -520, 0, 57, 560, 0, 0, 182, -512, -880, 0, 1190, -448, 884, 0, 0, 0, -1400, 0, -1330, 1000, 1820, 0, -646, 1280, 0, 0, -1331, -2464, 380, 0, 1120, 0, 2576, 0, 0, -880, 1748, 0, -3850, 0, -3400, 0, 2703, 4160, -2500, 0, 3458

Offset: 0

N. J. A. Sloane

Number 22 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_4(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 9 form of weight 4.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 8*x + 20*x^2 - 70*x^3 + 64*x^4 + 56*x^5 - 125*x^6 - 160*x^7 + ...
G.f. = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Shaun Cooper, Minchael D. Hirschhorn and Richard Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 319.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Morris Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A033687, A033690, A092342, A153728.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Basis( CuspForms( Gamma0(9), 4), 56) [1]; /* Michael Somos, Dec 09 2013 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8, {x, 0, n}]; (* Michael Somos, Sep 29 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^8, {x, 0, n}]; (* Michael Somos, Dec 09 2013 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(3*e/2)), forstep( y=sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x=-x); break)); a0=1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 23 2006 */

CuspForms( Gamma0(9), 4, prec=56).0; # Michael Somos, May 28 2013

Expansion of q^(-1/3) * eta(q)^8 in powers of q.
Expansion of q^(-1/3) * b(q)^3 * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Expansion of q^(-1) * b(q) * c(q)^3 / 27 in powers of q^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Euler transform of period 1 sequence [ -8, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-1)^(e/2) * p^(3*e/2) if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2))*p^3 if p == 1 (mod 3) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Michael Somos, Aug 23 2006
Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 - u * w * (u + 16 * w). - Michael Somos, Feb 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 29 2011
G.f.: Product_{k>0} (1 - x^k)^8.
a(2*n) = A153728(n). - Michael Somos, Sep 29 2011
a(4*n + 1) = -8 * a(n). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
A092342(n) = a(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{n>=0} a(n) * q^(3*n + 1) = (Sum_{i,j,k in Z} (i-j) * (j-k) * (k-i) * q^((i*i + j*j + k*k) / 2)) / 2 where 0 = i+j+k, i == 1 (mod 3), j == 2 (mod 3), and k == 0 (mod 3). - Michael Somos, Sep 22 2014
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-8*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 2 (mod 3). Then a( M^2*n + (M^2 - 1)/3 ) = (-1)^k*M^3*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p^3)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(3*e+3) - ((x-sqrt(-3)*y)/2)^(3*e+3))/(((x+sqrt(-3)*y)/2)^3 - ((x-sqrt(-3)*y)/2)^3) if p == 1 (mod 3) where 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Jianing Song, Mar 19 2022

Corrected by Charles R Greathouse IV, Sep 02 2009

A050470 a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.

Original entry on oeis.org

1, 4, 8, 16, 26, 32, 48, 64, 73, 104, 120, 128, 170, 192, 208, 256, 290, 292, 360, 416, 384, 480, 528, 512, 651, 680, 656, 768, 842, 832, 960, 1024, 960, 1160, 1248, 1168, 1370, 1440, 1360, 1664, 1682, 1536, 1848, 1920, 1898, 2112, 2208, 2048, 2353, 2604

Offset: 1

N. J. A. Sloane, Dec 23 1999

Number 7 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Multiplicative because it is the Dirichlet convolution of A000290 = n^2 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

			G.f. = q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A050461, A050465, A101455, A120030, A153071.
Cf. A027750, A000122, A000700, A010054, A121373.
Glaisher's E'_i (i=0..12): A002654, A050469, this sequence, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

a050470 n = a050461 n - a050465 n  -- Reinhard Zumkeller, Mar 06 2012

Basis( ModularForms( Gamma1(4), 3), 51) [2]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^3 (QPochhammer[ q^4] / QPochhammer[ q])^2)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^2 / 4)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[n/d, 2] (-1)^Quotient[n/d, 2], {d, Divisors@n}]]; (* Michael Somos, May 17 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(2*e+2) - s[p]^(e+1))/(p^2 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * (eta(x^4 + A) / eta(x + A))^4, n))}; /* Michael Somos, May 17 2015 */

from math import prod
from sympy import factorint
def A050470(n): return prod((p**(e+1<<1)-(m:=(0,1,0,-1)[p&3]))//(p**2-m) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

G.f.: Sum_{n>=1} n^2*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Aug 08 2005: (Start)
Euler transform of period 4 sequence [ 4, -2, 4, -6, ...].
Expansion of eta(q^2)^6 * eta(q^4)^4 / eta(q)^4 in powers of q.
G.f.: x Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 8*v) * (v - 4*w) - v^2 * (v - 8*w)^2. (End)
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k) / (1 - x^k)^3. - Michael Somos, Sep 02 2005
Expansion of q * phi(q)^2 * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120030.
a(n) = A050461(n) - A050465(n). - Reinhard Zumkeller, Mar 06 2012
Multiplicative with a(p^e) = ((p^2)^(e+1) - Chi(p)^(e+1))/(p^2 - Chi(p)), Chi = A101455. - Jianing Song, Oct 30 2019
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/32 (A153071). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^2*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

A002171 Glaisher's chi numbers. a(n) = chi(4*n + 1).

Original entry on oeis.org

1, -2, -3, 6, 2, 0, -1, -10, 0, -2, 10, 6, -7, 14, 0, -10, -12, 0, -6, 0, 9, -4, 10, 0, 18, -2, 0, 6, -14, -18, -11, 12, 0, 0, -22, 0, 20, 14, -6, 22, 0, 0, 23, -26, 0, -18, 4, 0, -14, -2, 0, -20, 0, 0, 0, 12, 3, 30, 26, 0, -30, 14, 0, 0, 2, 30, -28, -26, 0, -18, 10, 0, -13, -34, 0, 0, 20, 0, 26, 22, 0, -6, 0, 6, 18, 0

Offset: 0

N. J. A. Sloane

Number 49 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Glaisher (1884) essentially defines chi(n) as the sum over all solutions of n = x^2 + y^2 with even y and nonnegative odd x of x * (-1)^((x + y - 1)/2) and proves that it is multiplicative. If n is not == 1 (mod 4) then chi(n) = 0. - Michael Somos, Jun 18 2012
Denoted by g_2(q) in Cynk and Hulek on page 8 as the unique weight 2 level 32 newform. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012
The weight 2 level N = 32 newform (eta(q^4)*eta(q^8))^2 belongs to the elliptic curves y^2 = x^3 + 4*x , y^2 = x^3 - x, y^2 = x^3 - 11*x - 14 and y^2 = x^3 - 11*x + 14. See the Martin-Ono link, Theorem 2, row N = 32, and the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 26 2016

			G.f. = 1 - 2*x - 3*x^2 + 6*x^3 + 2*x^4 - x^6 - 10*x^7 - 2*x^9 + 10*x^10 + ...
G.f. (eta(q^4)*eta(q^8))^2 = q - 2*q^5 - 3*q^9 + 6*q^13 + 2*q^17 - q^25 - 10*q^29 - 2*q^37 + 10*q^41 + ...

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)
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
J. E. Cremona, Algorithms for Modular Elliptic Curves.
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
T. Ishikawa, Congruences between binomial coefficients binomial(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
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 sequences related to Glaisher's numbers

Cf. A000203, A002172, A278720, A279955.

A := Basis( ModularForms( Gamma0(32), 2), 341); A[2] - 2*A[6]; /* Michael Somos, Jun 12 2014 */

qEigenform( EllipticCurve( [0, 0, 0, -1, 0]), 341); /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(32), 2), 341)[1]; /* Michael Somos, Mar 25 2015 */

max=100; f[x_] := Product[(1-x^k)*(1-x^(2k)), {k, 1, max}]^2; CoefficientList[ Series[ f[x], {x, 0, max}], x](* Jean-François Alcover, Jan 04 2012, after g.f. *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x^2]^3, {x, 0, n}]; (* Michael Somos, Jun 18 2012 *)

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, -1, 0], 1), 4*n + 1))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^2, n))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor( 4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, (-p)^(e/2) * (1 + (-1)^e) / 2, forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0 = 1; y = a1 = x * (-1)^((x + y)\2) * if(y, 2, 1); for(i=2, e, x = y * a1 - p * a0; a0=a1; a1=x); a1 )))}; /* Michael Somos, Jun 18 2012 */

Expansion of (psi(x) * phi(-x))^2 = phi(-x) * f(-x^2)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q)^2 * eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [-2, -4, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative with b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) and b(p) = p - number of solutions of y^2 = x^3 - x (mod p). - Michael Somos, Jul 27 2006. b(p(n)) = A278720(n). - Wolfdieter Lang, Dec 26 2016
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^2.
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (Sum_{k>=0} (-1)^k * (2*k + 1) * x^(2*k + 1)^2) * (Sum_{k in Z} (-1)^k * x^(4*k)^2).
Coefficients of L-series for elliptic curve "32a2": y^2 = x^3 - x.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

Original entry on oeis.org

1, -6, 9, 10, -30, 0, 11, 42, 0, -70, 18, -54, 49, 90, 0, -22, -60, 0, -110, 0, 81, 180, -78, 0, 130, -198, 0, -182, -30, 90, 121, 84, 0, 0, 210, 0, -252, -102, -270, 170, 0, 0, -69, 330, 0, -38, 420, 0, -190, -390, 0, -108, 0, 0, 0, -300, 99, 442, 210, 0, 418, -294, 0, 0, -510, 378, -540, 138, 0

Offset: 0

N. J. A. Sloane

This is Glaisher's function lambda(m). It appears to be defined only for odd m, and lambda(4t-1) = 0 (t >= 1), lambda(4t+1) = a(t) (t >= 0). - N. J. A. Sloane, Nov 25 2018
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 36 of the 74 eta-quotients listed in Table I of Martin (1996).
Dickson, v.2, p. 295 briefly states a result of Glaisher, 1883, pp 212-215. This result is that a(n) is the sum over all solutions of 16*n + 4 = x^2 + y^2 + z^2 + w^2 in nonnegative odd integers of chi(x) and is also the sum over all solutions of 8*n + 2 = x^2 + y^2 in nonnegative odd integers of chi(x) * chi(y) where chi(x) = x if x == 1 (mod 4) and -x if x == 3 (mod 4). [Michael Somos, Jun 18 2012]
Denoted by g_3(q) in Cynk and Hulek on page 8 as the unique weight 3 Hecke eigenform of level 16 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012

			G.f. = 1 - 6*x + 9*x^2 + 10*x^3 - 30*x^4 + 11*x^6 + 42*x^7 - 70*x^9 + 18*x^10 + ...
G.f. = q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 295, and vol. 3, p. 134.
J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See page 340.
J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m), Quart. J. Math, 37 (1906), 36-48.
Morris Newman, A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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
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. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
J. W. L. Glaisher, Note on the Compositions of a Number as a Sum of Two and Four Uneven Squares, Quarterly Journal of Pure and Applied Mathematics, 19 (1883), 212-215.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
S. Milne and V. Leininger, Some new infinite families of eta function identities, Methods and Applications of Analysis 6 (1999), 225--248.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
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
Index entries for sequences mentioned by Glaisher

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

A := Basis( ModularForms( Gamma1(16), 3), 274); A[2] - 6*A[6] + 9*A[10] + 10*A[14] - 30*A[18]; /* Michael Somos, May 17 2015 */

A := Basis( CuspForms( Gamma1(16), 3), 274); A[1] - 6*A[5]; /* Michael Somos, Jan 09 2017 */

a[ n_] := SeriesCoefficient[ 1/16 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q]^4 EllipticTheta[ 3, 0, q], {q, 0, 4 n + 1}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[ 16 n + 4]}, SeriesCoefficient[ Sum[ Mod[k, 2] q^k^2, {k, m}]^3 Sum[ KroneckerSymbol[ -4, k] k q^k^2, {k, m}], {q, 0, 16 n + 4}]]]; (* Michael Somos, Jun 12 2012 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^3 / (4 q^(1/2)), {q, 0, 2 n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/4) EllipticThetaPrime[ 1, -Pi/4, q] EllipticTheta[ 1, -Pi/4, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/16) EllipticThetaPrime[ 1, 0, q] EllipticTheta[ 1, -Pi/2, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 21 2006 */

{a(n)=local(tn=(sqrtint(8*n+1)+1)\2);polcoeff(sum(m=0,tn,(1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0,tn,sum(k=1,tn,(1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))),n)} /* Paul D. Hanna, Mar 15 2010 */

Expansion of q^(-1/4)/16 * theta_2(q)^4 * theta_3(q) * theta_4(q) in powers of q. - [Dickson, v. 3, p. 134] from Stieltjes footnote 160. Michael Somos, Jun 18 2012
Expansion of q^(-1/2) / 4 * k * k' * (K / (pi/2))^3 in powers of q^2 where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
G.f.: Product_{k>0}(1 - x^k)^6.
Given g.f. A(x), then A(q^4) = f(-q^4)^6 = phi(q) * phi(-q) * psi(q^2)^4 where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 23 2006
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^2 if p == 1 (mod 4) and b(p) = 2 * (x^2 - y^2) where p = x^2 + y^2 and y is even. - Michael Somos, Aug 23 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (1/2) * Sum_{u,v in Z} (u*u - 4*v*v) * x^(u*u + 4*v*v). - Michael Somos, Jun 14 2007
G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [Paul D. Hanna, Mar 15 2010]
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M be a positive integer whose prime factors are all congruent to 3 (mod 4) - see A004614. Then a( M^2*n + (M^2 - 1)/4 ) = M^2*a(n). See Cooper et al., equation 5. - Peter Bala, Dec 01 2020
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = ((x+y*i)^(2*e+2) - (x-y*i)^(2*e+2))/((x+y*i)^2 - (x-y*i)^2) if p == 1 (mod 4) where p = x^2 + y^2 and x is odd. - Jianing Song, Mar 19 2022

cp's OEIS Frontend

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

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A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

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A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

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A000727 Expansion of Product_{k >= 1} (1 - x^k)^4.

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A006571 Expansion of qProduct_{k>=1} (1-q^k)^2(1-q^(11*k))^2.