A116916 -id:A116916 - cp's OEIS frontend

A010816 Expansion of Product_{k>=1} (1 - x^k)^3.

1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -27, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

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

Also, number of different partitions of n into parts of -3 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

			G.f. = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + ...
G.f. for b(n): = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.14).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003, p. 285, Theorem 357 (Jacobi).
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410, Problem 23.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)

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. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
V. Kotesovec, The integration of q-series
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.
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. A000594, A116916, A258407.

# DedekindEta is defined in A000594.
A010816List(len) = DedekindEta(len, 3)
A010816List(39) |> println # Peter Luschny, Mar 10 2018

S:= series(mul(1-x^k,k=1..200)^3,x,201):
seq(coeff(S,x,j),j=0..200); # Robert Israel, Feb 01 2018
A010816 := n -> if issqr(8*n+1) then isqrt(8*n+1); (-1)^iquo(%, 2) * % else 0 fi:
seq(A010816(n), n=0..98); # Peter Luschny, Apr 17 2022

a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 22 2011 *)
a[ n_] := With[ {m = 8 n + 1}, If[m > 0 && OddQ[ Length @ Divisors @ m], Sqrt[m] KroneckerSymbol[-4, Sqrt[m]], 0]];  (* Michael Somos, Aug 26 2015 *)
CoefficientList[QPochhammer[q]^3 + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015 *)
a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[ x], (-1)^Quotient[ x, 2] x, 0]]; (* Michael Somos, Feb 01 2018 *)
a[ n_] := If[ n < 1, Boole[ n == 0], Times @@ (If[ # == 2 || OddQ[ #2], 0, (KroneckerSymbol[ -4, #] #)^(#2/2)] & @@@ FactorInteger[ 8 n + 1])]; (* Michael Somos, Feb 01 2018 *)

{a(n) = my(x); if( n<0, 0, if( issquare( 8*n + 1, &x), (-1)^(x\2) * x))}; /* Michael Somos, Nov 08 2005 */

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

from sympy import integer_nthroot
def A010816(n):
    a, b = integer_nthroot((n<<3)+1,2)
    return (-a if a&2 else a) if b else 0 # Chai Wah Wu, Nov 02 2024

G.f.: Product_{k>=1} (1-x^k)^3 = Sum_{n>=0} (-1)^n*(2*n+1)*x^(n*(n+1)/2) (Jacobi).
Given g.f. A(x), then q * A(q^8) = eta(q^8)^3 = theta_2(q^4)*theta_3*(q^4)*theta_4(q^4) / 2 = theta_1'(q^4) / (2*Pi). - Michael Somos, Nov 08 2005
Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.
a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(e/2) if p == 1 (mod 4), b(p^e) = (-p)^(e/2) if p == 3 (mod 4). - Michael Somos, Aug 22 2006
Expansion of f(-x)^3 in powers of x where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2^(9/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2007
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -3 * a(n). a(25*n + 3) = 5 * a(n). - Michael Somos, Sep 09 2007
a(3*n) = A116916(n).
a(n) = (t*(t+1)-2*n-1)*(t-r)*(-1)^(t+1), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 17 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
G.f.: Product_{n >= 1} (1 - q^(4*n))^3 * (1 + q^(4*n-1))^(-3) * (1 - q^(4*n-2))^6 * (1 + q^(4*n-3))^(-3). - Peter Bala, Jun 07 2025

A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -5, -7, 0, 0, 11, 0, -13, 0, 0, 0, 0, 17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0

Offset: 0

Michael Somos, Sep 08 2007

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
There is a plus sign on the left side and the first and third plus signs on the right side which should be minuses in Ramanujan's equation.

			G.f. = 1 - 5*x - 7*x^2 + 11*x^5 - 13*x^7 + 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.

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

Cf. A116916, A133089, A204850.

a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m (-1)^Boole[Mod[m, 8] > 4], 0]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 3 x QPochhammer[ -x^9]^3, {x, 0, 3 n}]; (* Michael Somos, Jun 19 2015 *)

{a(n) = if( issquare( 24*n + 1, &n), n * (-1) ^ (n%8 > 4), 0)};

{a(n) = my(A, p, e); if( n<0, 0, n = 24*n + 1; A = factor(n); prod(k = 1, matsize(A) [1], [p, e] = A[k,]; if( p < 5, 0, p *= kronecker( -2, p); if( e%2, 0, p^(e/2) ))))};

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

Expansion of f(x) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204850.
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
G.f.: Sum_{k in Z} Kronecker( 2, 2*k + 1) * (6*k + 1) * x^(k * (3*k + 1)/2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -5 * a(n). a(n) = (-1)^n * A116916(n).
a(n) = A133089(3*n) = A204850(3*n). - Michael Somos, Jun 19 2015

A282941 a(n) = A000730(7*n).

Original entry on oeis.org

1, 41, -176, 98, 322, -181, -140, -489, 112, 889, 14, -560, 125, 154, 756, -1317, -1778, 1554, -1218, 2688, 1764, -980, 71, -1575, 14, -1638, -419, 56, -1988, -2716, 6223, 6860, 1302, -700, -3416, -4733, -2548, -4725, 3836, 1106, 2631, 5096, -5656, 2660, -7875

Offset: 0

Seiichi Manyama, Feb 25 2017

sign

			G.f.: 1 + 41*q - 176*q^2 + 98*q^3 + 322*q^4 - 181*q^5 - 140*q^6 - 489*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 191.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000730, A116916, A282937, A282942.

G.f.: Product_{n>=1} (1 - q^n)^8/(1 - q^(7*n)) + 49*q*(Product_{n>=1} (1 - q^n)^4*(1 - q^(7*n))^3).
a(n) = (-1)^j mod 7 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 7.
a(n) = A282942(n) mod 49.

A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 0, 5, 0, 0, -7, 0, 0, 0, -18, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0

Offset: 0

Michael Somos, Dec 18 2011

sign

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

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

			G.f. = 1 + 6*x + 5*x^3 - 7*x^6 - 18*x^10 - 11*x^15 + 13*x^21 + 30*x^28 + ...
G.f. = q + 6*q^9 + 5*q^25 - 7*q^49 - 18*q^81 - 11*q^121 + 13*q^169 + 30*q^225 + ...

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. A010816, A116916.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3, {x, 0, n}]; (* Michael Somos, May 26 2014 *)

{a(n) = local(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6) * m * ((m%3 == 0) + 1), 0)};

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

Expansion of f(-x^3) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
Expansion of q^(-1/8) * (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 41472^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A116916.
G.f.: Sum_{k} -(-1)^k * (6*k - 1) * x^(3*k*(3*k - 1)/2) + Sum_{k>0} -(-1)^k * 6 * (2*k - 1) * x^(9*k*(k - 1)/2 + 1).
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A116916(n). a(9*n + 1) = 6 * A010816(n). a(25*n + 3) = 5 * a(n).
a(n) nonzero if and only if n is a triangular number.

A134756 Coefficients of a q-series of Zagier related to the Dedekind eta function.

Original entry on oeis.org

1, -5, -7, 0, 0, 11, 0, 13, 0, 0, 0, 0, -17, 0, 0, -19, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Nov 08 2007

sign

Obtained by formally "differentiating the Dedekind eta-function half a time".

			G.f. = 1 - 5*x - 7*x^2 + 11*x^5 + 13*x^7 - 17*x^12 - 19*x^15 + 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 + 13*q^169 - 17*q^289 - 19*q^361 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001). See Eq. (20).

Cf. A010815.

Apart from signs, same as A080332, A116916 and A133079. - N. J. A. Sloane, Nov 11 2007

a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ @ m, m KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Oct 15 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (If[ # < 5, 0, (1 - Mod[#2, 2]) (# KroneckerSymbol[ 12, #])^(#2/2)] & @@@ FactorInteger[ 24 n + 1])]; (* Michael Somos, Oct 15 2015 *)
s = QPochhammer[q] + O[q]^100; A010815 = CoefficientList[s, q]; nn = Range[0, Length[A010815]-1]; A134756 = Sqrt[24*nn+1]*A010815 (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( issquare( 24*n+1, &n), n * kronecker( 12, n), 0)};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n+1); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, (kronecker( 12, p) * p)^(e\2))))};

a(n) = b(24*n + 1) where b() is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 11 (mod 12), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 12).
G.f.: Sum_{k>0} Kronecker(12, k) * k * x^((k^2 - 1) / 24).
a(n) = sqrt(24*n + 1) * A010815(n).

A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.

Original entry on oeis.org

1, 5, 7, 0, 0, 11, 0, -13, 0, 0, 0, 0, -17, 0, 0, -19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jun 21 2010

sign

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

			G.f. = 1 + 5*x + 7*x^2 + 11*x^5 - 13*x^7 - 17*x^12 - 19*x^15 - 23*x^22 + ...
G.f. = q + 5*q^25 + 7*q^49 + 11*q^121 - 13*q^169 - 17*q^289 - 19*q^361 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
G. Köhler, Some eta-identities arising from theta series, Math. Scand. 66 (1990), 147-154.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Apart from signs, same as A080332, A116916, A133079 and A134756.

A178902[n_] := SeriesCoefficient[(QPochhammer[-q, -q]/QPochhammer[q, -q])^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A178902[n], {n, 0, 50}] (* G. C. Greubel, Aug 17 2017 *)
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m KroneckerSymbol[ -6, m], 0]]; (* Michael Somos, Apr 27 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^13 / (QPochhammer[ x] QPochhammer[ x^4])^5, {x, 0, n}]; (* Michael Somos, Apr 27 2018 *)

{a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -6, n), 0)};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, if( p%24<12, p, -p)^(e\2))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 / (eta(x + A)^5 * eta(x^4 + A)^5), n))};

Expansion of f(q) * phi(q)^2 = f(q)^3 * chi(q)^2 = phi(q)^3 / chi(q) in powers of q where f(), phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [5, -8, 5, -3, ...].
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 + x^(2*k - 1))^5 = Sum_{k>0} Kronecker( -6, k) * k * x^((k^2 - 1) / 24) = Sum_{k in Z} (6*k + 1) * (-1)^floor(k/2) * x^(k * (3*k + 1) / 2).
a(n) = (-1)^n * A080332(n).

cp's OEIS Frontend

A010816 Expansion of Product_{k>=1} (1 - x^k)^3.

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A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.

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A282941 a(n) = A000730(7*n).

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A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

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A134756 Coefficients of a q-series of Zagier related to the Dedekind eta function.

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A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.

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