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A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.
(Formerly M4076 N1691)
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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.

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).

LINKS

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. 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]

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

M. 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

FORMULA

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) = (-1)^y * (x^2 - y^2) where p = x^2 + y^2. - 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

EXAMPLE

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 + ...

MATHEMATICA

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 *)

PROG

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

(PARI) {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), for( i=1, sqrtint(p\2), if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2) * (-1)^y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 21 2006 */

(PARI) {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 */

(MAGMA) 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 */

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

CROSSREFS

Sequence in context: A051221 A029843 A209941 * A280666 A282937 A106248

Adjacent sequences:  A000726 A000727 A000728 * A000730 A000731 A000732

KEYWORD

easy,sign

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 16 05:13 EST 2018. Contains 317257 sequences. (Running on oeis4.)