A138506 -id:A138506 - cp's OEIS frontend

A109064 Expansion of eta(q)^5 / eta(q^5) in powers of q.

1, -5, 5, 10, -15, -5, -10, 30, 25, -35, 5, -60, 30, 60, -30, 10, -55, 80, 35, -100, -15, -60, 60, 110, -50, -5, -60, 100, 90, -150, -10, -160, 105, 120, -80, 30, -105, 180, 100, -120, 25, -210, 60, 210, -180, -35, -110, 230, 110, -215, 5, -160, 180, 260

Offset: 0

Michael Somos, Jun 17 2005

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Number 12 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 5*q + 5*q^2 + 10*q^3 - 15*q^4 - 5*q^5 - 10*q^6 + 30*q^7 + 25*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
William Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 151.
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.
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression B^5/C in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]

Cf. A053723, A109091, A138506, A277212, A285896.

A := Basis( ModularForms( Gamma1(5), 2), 54); A[1] - 5*A[2] + 5*A[3]; /* Michael Somos, May 19 2015 */

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 5)=0, -4, -5), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 07 2017

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^5 / QPochhammer[ q^5], {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -5 DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* Michael Somos, May 19 2015 *)

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

{a(n) = if( n<1, n==0, -5 * sumdiv(n, d, d * kronecker(5, d)))}; /* Michael Somos, May 19 2015 */

Euler transform of period 5 sequence [ -5, -5, -5, -5, -4, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 2 * u*v*w + u^2*w - 4 * u*w^2.
a(n) = -5 * b(n) where b() is multiplicative with a(0) = 1, b(p^e) = 1 if p=5,  b(p^e) = b(p) * b(p^(e-1)) - Kronecker(5, p) * p * b(p^(e-2)) otherwise. - Michael Somos, May 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(5/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053723. - Michael Somos, May 19 2015
G.f.: Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k)).
a(n) = -5 * A109091(n), unless n=0. a(n) = (-1)^n * A138506(n). a(5*n) = a(n).
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - Amiram Eldar, Jan 29 2024

A138512 Expansion of q * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 2, -3, 5, -2, 6, -5, 7, -5, 12, -6, 12, -6, 10, -11, 16, -7, 20, -15, 12, -12, 22, -10, 25, -12, 20, -18, 30, -10, 32, -21, 24, -16, 30, -21, 36, -20, 24, -25, 42, -12, 42, -36, 35, -22, 46, -22, 43, -25, 32, -36, 52, -20, 60, -30, 40, -30, 60, -30, 62

Offset: 1

Michael Somos, Mar 21 2008

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

			G.f. = q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 2*q^6 + 6*q^7 - 5*q^8 + 7*q^9 - 5*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A053723, A134080, A138506, A328717.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5]^5 / QPochhammer[ -q], {q, 0, n}]; (* Michael Somos, Sep 25 2015 *)
a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ n/d KroneckerSymbol[ 5, d], { d, Divisors @ n}]]; (* Michael Somos, Sep 25 2015 *)

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, n/d)))};

{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(2^(e+1) - (-1)^(e+1)) / 3, f = kronecker(5, p); (p^(e+1) - f^(e+1)) / (p - f) ))) };

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^15 / (eta(x^2 + A)^3 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};

Expansion of eta(q) * eta(q^4) * eta(q^10)^15 / (eta(q^2)^3 * eta(q^5)^5 * eta(q^20)^5) in powers of q.
Euler transform of period 20 sequence [ -1, 2, -1, 1, 4, 2, -1, 1, -1, -8, -1, 1, -1, 2, 4, 1, -1, 2, -1, -4, ...].
a(n) is multiplicative with a(2^e) = -(2^(e+1) - (-1)^(e+1)) / 3 if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
a(n) = -(-1)^n * A053723(n-1).
From Michael Somos, Sep 25 2015: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (16/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138506.
a(2*n + 1) = A134080(n). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A328717 / 8 = 0.0882764... . - Amiram Eldar, Nov 23 2023

A138507 Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, -3, 1, -2, -6, 5, 7, 1, 12, 6, -12, -6, -2, -11, -16, 7, 20, -3, 12, 12, -22, -10, 1, -12, -20, 18, 30, -2, 32, 21, -24, -16, -6, -21, -36, 20, 24, 5, 42, 12, -42, -36, 7, -22, -46, 22, 43, 1, 32, 36, -52, -20, 12, -30, -40, 30, 60, 6, 62, 32, -42, -43, -12, -24, -66, 48, 44, -6, 72, 35, -72, -36, -2, -60, -72, 24

Offset: 1

Michael Somos, Mar 21 2008

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			q + q^2 - 2*q^3 - 3*q^4 + q^5 - 2*q^6 - 6*q^7 + 5*q^8 + 7*q^9 + ...

Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

-(-1)^n * A109091(n) = a(n). A138506(n) = 5 * a(n) unless n=0.

Cf. A000122, A000700, A010054, A121373.

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, d)))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(-x + A)^5 / eta(-x^5 + A) - 1) / 5, n))}

a(n) is multiplicative with a(2^e) = ((-2)^(e+1) - 1) / 3, a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10), a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10).
G.f.: (Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)) - 1) / 5.
L.g.f.: log(1/(1 - x/(1 + x^2/(1 - x^3/(1 + x^4/(1 - x^5/(1 + ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - Amiram Eldar, Jan 29 2024

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A109064 Expansion of eta(q)^5 / eta(q^5) in powers of q.

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A138512 Expansion of q * f(q^5)^5 / f(q) in powers of q where f() is a Ramanujan theta function.

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A138507 Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.

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