A129391 -id:A129391 - cp's OEIS frontend

A129390 Expansion of phi(x) * phi(-x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 2, 1, 2, 3, 0, 0, 2, 0, 0, 4, 2, 1, 4, 2, 0, 0, 2, 0, 0, 2, 2, 3, 2, 3, 0, 0, 0, 0, 0, 2, 6, 0, 2, 4, 0, 0, 2, 0, 0, 5, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 4, 2, 2, 0, 0, 2, 0, 0, 1, 4, 1, 2, 4, 0, 0, 4, 0, 0, 4, 0, 2, 6, 2, 0, 0, 0, 0, 0, 4, 2, 0, 2, 1, 0, 0

Offset: 0

Michael Somos, Apr 13 2007

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

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^7 + 4*x^10 + 2*x^11 + x^12 + 4*x^13 + ...
G.f. = q + 2*q^3 + q^5 + 2*q^7 + 3*q^9 + 2*q^15 + 4*q^21 + 2*q^23 + q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035710, A129391.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -20, #]&]]; (* Michael Somos, Nov 12 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( -20, d)))};

{a(n) = my(A, p, e); if(n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p%20 <10, e+1, 1-e%2) ))};

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

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^5)^2 * eta(q^20) / (eta(q)^2 * eta(q^4) * eta(q^10)^2) in powers of q.
Euler transform of period 20 sequence [ 2, -2, 2, -1, 0, -2, 2, -1, 2, -2, 2, -1, 2, -2, 0, -1, 2, -2, 2, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), b(p^e) = (1 + (-1)^e)/2 if p == 11, 13, 17, 19 (mod 20).
G.f.: Sum_{k>0} a(k) * x^(2*k - 1) = Sum_{k>0} f(x^(2*k - 1)) where f(x) := x * (1 + x^2) * (1 + x^6) / (1 + x^10).
a(n) = (-1)^n * A129391(n). a(n) = A035710(2*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Dec 28 2023

A111949 Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -1, -2, 1, 1, 2, -2, -1, 3, -1, 0, -2, 0, 2, -2, 1, 0, -3, 0, 1, 4, 0, -2, 2, 1, 0, -4, -2, 2, 2, 0, -1, 0, 0, -2, 3, 0, 0, 0, -1, 2, -4, -2, 0, 3, 2, -2, -2, 3, -1, 0, 0, 0, 4, 0, 2, 0, -2, 0, -2, 2, 0, -6, 1, 0, 0, -2, 0, 4, 2, 0, -3, 0, 0, -2, 0, 0, 0, 0, 1, 5, -2, -2, 4, 0, 2, -4, 0, 2, -3, 0, -2, 0, 2, 0, 2, 0, -3, 0, 1, 2, 0, -2, 0, 4

Offset: 1

Michael Somos, Aug 22 2005

Number 37 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Y. 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

Cf. A033764, A035170, A129391.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^10] QPochhammer[ q^20] / (QPochhammer[ q^4] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[d, 2] (-1)^Quotient[d, 2] KroneckerSymbol[ n/d, 5], { d, Divisors[ n]}]]; (* Michael Somos, May 19 2015 *)

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

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

{a(n) = if( n<1, 0, qfrep( [1, 0; 0, 5], n)[n] - qfrep( [2, 1; 1, 3], n)[n])};

Euler transform of period 20 sequence [-1, -2, -1, -1, 0, -2, -1, -1, -1, -2, -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, ...].
a(n) is multiplicative with a(p^e) = (-1)^e if p = 2, a(p^e) = 1 if p = 5, a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e + 1 if p == 1, 9 (mod 20), a(p^e) = (e + 1)*(-1)^e if p == 3, 7 (mod 20).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)).
G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^(2*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 - x^(20*k)) / (1 + x^(2*k)).
|a(n)| = A035170(n). a(2*n) = -a(n). a(2*n + 1) = A129391(n). a(4*n + 3) = -2 * A033764(n).
a(5*n) = a(n). - Michael Somos, May 19 2015

A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.

Original entry on oeis.org

1, 1, -2, -1, 1, -2, -2, 1, 3, 1, 0, 2, 0, -2, -2, -1, 0, 3, 0, -1, 4, 0, -2, -2, 1, 0, -4, 2, 2, -2, 0, 1, 0, 0, -2, -3, 0, 0, 0, 1, 2, 4, -2, 0, 3, -2, -2, 2, 3, 1, 0, 0, 0, -4, 0, -2, 0, 2, 0, 2, 2, 0, -6, -1, 0, 0, -2, 0, 4, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, -1, 5, 2, -2, -4, 0, -2, -4, 0, 2, 3, 0, 2, 0, -2, 0, -2, 0, 3, 0, -1, 2, 0, -2, 0, 4

Offset: 1

Michael Somos, Aug 07 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 - q^4 + q^5 - 2*q^6 - 2*q^7 + q^8 + 3*q^9 + q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 338, Eq. (3.22), p. 342, Eq. (3.41).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A111949, A129391.

a[ n_] := SeriesCoefficient[ 2 q^(13/8) EllipticTheta[ 4, 0, q^2]^2 QPochhammer[ q^20]^2 / ( QPochhammer[ q] EllipticTheta[ 2, 0, q^(5/2)]), {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)

{a(n) = if( n<1, 0, (-1)^n * (qfrep([2, 1; 1, 3], n)[n] - qfrep([1, 0; 0, 5], n)[n] ))};

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

Expansion of q * phi(-q^2) * chi(q) * psi(q^10) * chi(-q^5) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q^2)^2 * psi(-q^5)^2 / (f(-q) * f(-q^5)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015
Euler transform of period 20 sequence [ 1, -3, 1, -1, 0, -3, 1, -1, 1, -2, 1, -1, 1, -3, 0, -1, 1, -3, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e unless e=0, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A129391.
G.f.: Sum_{k>0} -(-1)^k F(x^(2*k - 1)) where F(x) = x * (1 + x) * (1 - x^2) / (1 + x^5).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2 * (1 - x^(5*k)) * ( 1 + x^(10*k))^2.
a(n) = -(-1)^n * A111949(n).

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A129390 Expansion of phi(x) * phi(-x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions.

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A111949 Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.

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A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.

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