A113185 -id:A113185 - cp's OEIS frontend

A138483 Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 3, 2, 1, 5, 6, 6, 7, 7, 15, 12, 2, 12, 18, 10, 9, 16, 21, 20, 5, 12, 36, 22, 14, 25, 36, 20, 6, 30, 30, 32, 23, 24, 48, 30, 7, 36, 60, 24, 35, 42, 36, 42, 12, 35, 66, 46, 18, 43, 75, 32, 12, 52, 60, 60, 42, 40, 90, 60, 10, 62, 96, 42, 41, 60, 72, 66, 16, 44

Offset: 1

Michael Somos, Mar 20 2008

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

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A110712, A113185.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 25, #] (-1)^Quotient[# + 2, 5] If[ Mod[#, 4] > 0, 1, 5] &]]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ q EllipticTheta[3, 0, q] QPochhammer[ -q, q^2] QPochhammer[ -q^5] QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^3 EllipticTheta[3, 0, q^5] - EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5]^3) / 4, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))};

{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) - 5*(-1)^e) / 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^2 + A)^7 * eta(x^10 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};

Expansion of q * phi(q) * chi(q) * f(q^5) * f(-q^10)^2 in powers of q where phi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^7 * eta(q^10)^5 / (eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [3, -4, 3, -1, 4, -4, 3, -1, 3, -8, 3, -1, 3, -4, 4, -1, 3, -4, 3, -4, ...].
a(n) is multiplicative with a(2^e) = (2^(e+1) - 5*(-1)^e) / 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).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113185.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^3 * (1 - x^(5*k))^3 * (1 + x^(10*k-5))^4 * (1 + x^(10*k))^3.
a(n) = -(-1)^n * A110712(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - Amiram Eldar, Nov 24 2023

A132069 Expansion of eta(q) * eta(q^2)^2 * eta(q^5)^3 / eta(q^10)^2 in powers of q.

Original entry on oeis.org

1, -1, -3, 2, 1, -1, 6, 6, -7, -7, -3, -12, -2, 12, 18, 2, 9, 16, -21, -20, 1, -12, -36, 22, 14, -1, 36, 20, -6, -30, 6, -32, -23, 24, 48, 6, 7, 36, -60, -24, -7, -42, -36, 42, 12, -7, 66, 46, -18, -43, -3, -32, -12, 52, 60, -12, 42, 40, -90, -60, -2, -62, -96, 42, 41, 12, 72, 66, -16, -44, 18, -72, -49, 72, 108, 2, 20, 72

Offset: 0

Michael Somos, Aug 08 2007, Mar 20 2008

sign

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

Denoted by z(q) = q d/dq log k(q) in Cooper (2009) where k() is the g.f. of A112274. - Michael Somos, Jul 08 2012

			G.f. = 1 - q - 3*q^2 + 2*q^3 + q^4 - q^5 + 6*q^6 + 6*q^7 - 7*q^8 - 7*q^9 +...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 253 Eq. (8.12)

G. C. Greubel, Table of n, a(n) for n = 0..10000
Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 312, eq. (1.4).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A112274, A113185, A129303.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ 5, #] # (-1)^# &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^5]^3 / QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5])/4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

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

{a(n) = my(A, p, e, a1); if( n<1, n==0, A = factor(n); -prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p>2, p *= kronecker(5, p); (p^(e+1) - 1) / (p - 1), (5 + (-2)^(e+1)) / 3)))};

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

Expansion of (5 * phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 10 sequence [-1, -3, -1, -3, -4, -3, -1, -3, -1, -4, ...].
a(n) = -b(n) where b() is multiplicative with b(5^e) = 1, b(2^e) = 2 - ((-2)^(e+1) - 1) / (-2 - 1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10).
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k))^2 * (1 - x^(5*k)) / (1 + x^(5*k))^2.
G.f.: 1 + Sum_{k>0} (-1)^k * k * x^k / (1 - x^k) * Kronecker(5, k).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129303.
a(n) = (-1)^n * A113185(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - Amiram Eldar, Jan 28 2024

A259030 a(n) is multiplicative with a(2^e) = -(1 - (-1)^e) / 2 if e > 0, a(p^e) = Kronecker(5, p)^e if p > 2.

Original entry on oeis.org

1, -1, -1, 0, 0, 1, -1, -1, 1, 0, 1, 0, -1, 1, 0, 0, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, -1

Offset: 1

Michael Somos, Jun 17 2015

			G.f. = x - x^2 - x^3 + x^6 - x^7 - x^8 + x^9 + x^11 - x^13 + x^14 - x^17 + ...

Cf. A080891, A105368, A113185.

f[p_, e_] := KroneckerSymbol[5, p]^e; f[2, e_] := -(1 - (-1)^e) / 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

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

a(n) = a(-n) for all n in Z.
a(2*n + 1) = A105368(n). a(4*n + 1) = A080891(n-1). a(4*n + 2) = - A105368(n). a(4*n - 1) = A080891(n+1).
A113185(n) = Sum_{d|n} d * a(d) * -(-1)^(n/d) if n > 0.
G.f.: f(x) - Sum_{k>0} f(x^2^(2*k-1)) where f(x) := x * (1 - x^2) * (1 - x^6) / (1 - x^10).

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A138483 Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.

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A132069 Expansion of eta(q) * eta(q^2)^2 * eta(q^5)^3 / eta(q^10)^2 in powers of q.

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A259030 a(n) is multiplicative with a(2^e) = -(1 - (-1)^e) / 2 if e > 0, a(p^e) = Kronecker(5, p)^e if p > 2.

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