A030206 -id:A030206 - cp's OEIS frontend

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

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

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins, May 13 2008
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12 (but not explicitly listed).
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
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + 2*x^12 + 2*x^14 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Johnathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 697.
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
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.
Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From _Jeremy Lovejoy_, Jun 12 2009]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000731, A002324, A005882, A006938, A033685.

Cf. A045831, A053723, A081622.

Basis( ModularForms( Gamma1(9), 1), 316) [2]; /* Michael Somos, May 06 2015 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Sep 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Sep 01 2015 *)

{a(n) = if( n<0, 0, sumdiv( 3*n + 1, d, kronecker( -3, d)))}; /* Michael Somos, Nov 03 2005 */

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor( 3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%6==1, e+1, 1-e%2)))}; /* Michael Somos, May 06 2015 */

Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of q^(-1/3) * eta(q^3)^3 / eta(q) in powers of q.
a(4*n + 1) = a(n). - Michael Somos, Dec 06 2004
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^e) = 0^e if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q)= q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3. - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. - Michael Somos, May 20 2005
G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k).
G.f.: Sum_{k in Z} x^k / (1 - x^(3*k + 1)) = Sum_{k in Z} x^k / (1 - x^(6*k + 2)). - Michael Somos, Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A005928.
a(n) = Sum_{d|3n+1} LegendreSymbol{d,3} - Brian Hopkins, May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy, Jun 12 2009]
a(n) = A002324(3*n + 1). 3*a(n) = A005882(n) = A033685(3*n + 1). - Michael Somos, Apr 04 2003
G.f.: (2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5)) / 3 where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/3^(3/2). - Vaclav Kotesovec, Dec 17 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

A130539 Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 4, -13, 0, -1, 16, 11, 0, 25, -52, -46, 0, 47, 0, -22, 0, 120, -4, 0, 0, -121, 64, -109, 0, -97, 44, 131, 0, 0, 0, 13, 0, 167, 100, -37, 0, -214, -208, 0, 0, 121, -184, 146, 0, -143, 0, 251, 0, 0, 188, 59, 0, -118, 0, 299, 0, -168, -88, -325, 0, -313

Offset: 0

Michael Somos, Jun 03 2007

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Denoted by g_3(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 27 form of weight 3.
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.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

			G.f. = 1 + 4*x - 13*x^2 - x^4 + 16*x^5 + 11*x^6 + 25*x^8 - 52*x^9 - 46*x^10 + ...
G.f. = q + 4*q^4 - 13*q^7 - q^13 + 16*q^16 + 11*q^19 + 25*q^25 - 52*q^28 - ...

S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds

Cf. A000731, A030206, A033687.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^3] (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3), {x, 0, n}]; (* Michael Somos, Oct 20 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, if( p%3==2, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x=y*a1 - p^2*a0; a0=a1; a1=x); a1))))};

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

Expansion of q^(-1/3) * ( eta(q)^5 * eta(q^3) + 9 * eta(q)^2 * eta(q^3) * eta(q^9)^3 ) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = x^2 - 2*p, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3).
G.f.: Sum_{k>=0} a(k) * x^(3*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 7*v*v) * x^(u*u + u*v + 7*v*v). - Michael Somos, Jun 14 2007
a(4*n + 1) = 4*a(n). a(4*n + 3) = 0. - Michael Somos, Oct 20 2015

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

Original entry on oeis.org

1, -1, -2, -2, 3, 8, 0, -2, -10, -4, 2, 4, 10, -8, -4, 0, 7, 12, 4, -2, -16, -16, 4, 8, 0, -7, -4, -2, 10, 24, 8, -2, -26, 0, 2, 8, 12, -16, -8, -8, 10, 12, 0, -6, -20, -16, 4, 8, 26, -7, -10, 0, 16, 40, 0, -4, -20, -24, 6, 4, 0, -16, -12, -8, 15, 24, 8, -6

Offset: 0

Michael Somos, Nov 26 2019

sign

			G.f. = 1 - x - 2*x^2 - 2*x^3 + 3*x^4 + 8*x^5 - 2*x^7 - 10*x^8 - 4*x^9 + ...

Cf. A028967, A030206, A195848, A224822, A329956, A329957.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^2, {x, 0, n}];

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

Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].
G.f.: Product_{k>=1} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k))^2.
Convolution of A030206 and A195848.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.
a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

A279005 Expansion of 3 * q * b(q^9)^3 / c(q^3) in powers of q^3 where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -1, -1, -6, 6, 7, 9, -8, -15, 13, -19, -4, -49, 57, 61, 32, -14, -75, 47, -98, -23, -124, 130, 152, 116, -37, -182, 96, -168, 0, -335, 352, 342, 196, -117, -390, 230, -440, -107, -600, 637, 671, 480, -184, -704, 469, -727, -112, -1235, 1241, 1291, 722, -341

Offset: 0

Michael Somos, Dec 10 2016

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - x - x^2 - 6*x^3 + 6*x^4 + 7*x^5 + 9*x^6 - 8*x^7 - 15*x^8 + ...
G.f. = q^-1 - q^2 - q^5 - 6*q^8 + 6*q^11 + 7*q^14 + 9*q^17 - 8*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)

Cf. A030206.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3]^6 / QPochhammer[ x^9]^3, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A)^6 / eta(x^9 + A)^3, n))};

Expansion of q * eta(q^3) * eta(q^9)^6 / eta(q^27)^3 in powers of q^3.
Euler transform of period 9 sequence [ -1, -1, -7, -1, -1, -7, -1, -1, -4, ...].
a(5*n + 2) / a(2) == A030206(n) (mod 5). a(125*n + 42) / a(42) == A030206(n) (mod 25). [Amanda Clemm, 2016]

A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.

Original entry on oeis.org

1, -3, 0, 3, 6, 0, -18, 3, 0, 12, 0, 0, 21, -15, 0, -36, -12, 0, 36, 21, 0, 24, 0, 0, -90, 15, 0, 12, -6, 0, 54, 12, 0, -72, 0, 0, 84, -33, 0, 42, 0, 0, -144, -24, 0, 72, 0, 0, 93, 18, 0, -108, 30, 0, 36, 0, 0, 60, 0, 0, -252, 3, 0, 96, 24, 0, 108, -15, 0

Offset: 0

Michael Somos, Feb 19 2017

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

G.f. is a period 1 Fourier series which satisfies f(-1 / (9*t)) = 729 (t/i)^2 g(t) where g() is the g.f. for A282611.

			G.f. = 1 - 3*q + 3*q^3 + 6*q^4 - 18*q^6 + 3*q^7 + 12*q^9 + 21*q^12 - 15*q^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A030206, A281722, A282611.

A := Basis( ModularForms( Gamma0(27), 2), 69); A[1] - 3*A[2] + 3*A[4] + 6*A[5] - 18*A[6];

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^3]^2 / QPochhammer[ q^9], {q, 0, n}];

first(n)=my(q='x+O('x^(n+1))); Vec(eta(q)^3 * eta(q^3)^2 / eta(q^9)) \\ Charles R Greathouse IV, Jun 02 2017

Expansion of eta(q)^3 * eta(q^3)^2 / eta(q^9) in powers of q.
Euler transform of period 9 sequence [-3, -3, -5, -3, -3, -5, -3, -3, -4, ...].
a(3*n) = A281722(n). a(3*n + 1) = -3 * A030206(n). a(3*n + 2) = 0.

A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 143, 193, 266, 368, 493, 659, 892, 1170, 1543, 2035, 2642, 3422, 4448, 5694, 7294, 9334, 11839, 14982, 18968, 23812, 29868, 37410, 46598, 57924, 71953, 88913, 109728, 135212, 165991, 203407, 248986, 303706, 369939, 449967, 545820, 661038, 799629

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000041 and A035377.
Convolution of A000712 and A137569.
Convolution inverse of A030203.
Number of partitions of n if there are 2 kinds of parts that are multiples of 3.

			a(4) = 6 because we have [4], [3, 1], [3', 1], [2, 2], [2, 1, 1] and [1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000041, A000712, A002513, A030203, A030206, A035377, A112175, A137569, A318027, A318028.

a:=series(mul(1/((1-x^k)*(1-x^(3*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(3 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^3]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + 2 x^(2 k))/(k (1 - x^(3 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 3 k], {k, 0, n/3}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + 2*x^(2*k))/(k*(1 - x^(3*k)))).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (3 * 2^(5/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A258724 Expansion of f(-x)^11 / f(-x^3) + 27 * x * f(-x^3)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 16, 71, 0, -337, 256, -601, 0, 625, 1136, 194, 0, -529, 0, -3214, 0, 2640, -5392, 0, 0, 7199, 4096, 2903, 0, -1249, -9616, 4679, 0, 0, 0, -23927, 0, 9071, 10000, -19849, 0, 22034, 18176, 0, 0, 14641, 3104, -10942, 0, -42671, 0, 24359, 0, 0, -8464, -42121

Offset: 0

Michael Somos, Jun 08 2015

sign

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

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.

			G.f. = 1 + 16*x + 71*x^2 - 337*x^4 + 256*x^5 - 601*x^6 + 625*x^8 + ...
G.f. = q + 16*q^4 + 71*q^7 - 337*q^13 + 256*q^16 - 601*q^19 + ...

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. A000731, A030206, A033687, A130539.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^11 / QPochhammer[ x^3] + 27 x QPochhammer[ x^3]^11 / QPochhammer[ x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^11 / eta(x^3 + A) + 27 * x * eta(x^3 + A)^11 / eta(x + A), n))};

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = (y^2 - 2*p)^2 - 2*p^2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^4*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/3) * (eta(q)^11 / eta(q^3) + 27 * eta(q^3)^11 / eta(q)) in powers of q.
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^(2*e) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = (y^2 - 2*p)^2 - 2*p^2, 4*p = y^2 + 27*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t).
a(4*n + 3) = 0.

cp's OEIS Frontend

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

Extensions

A130539 Expansion of q^(-1/3) * a(q) * b(q) * c(q) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A279005 Expansion of 3 * q * b(q^9)^3 / c(q^3) in powers of q^3 where b(), c() are cubic AGM theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).