A138557 -id:A138557 - cp's OEIS frontend

A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function.

1, 2, 5, 6, 7, 12, 12, 10, 16, 20, 12, 22, 25, 20, 30, 32, 24, 30, 36, 24, 42, 42, 35, 46, 43, 32, 52, 60, 40, 60, 62, 42, 60, 66, 44, 72, 72, 50, 72, 80, 61, 82, 80, 60, 90, 72, 64, 100, 96, 84, 102, 102, 60, 106, 110, 72, 112, 110, 84, 96, 133, 84, 125, 126

Offset: 0

Michael Somos, Oct 07 2007

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

			G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 7*x^4 + 12*x^5 + 12*x^6 + 10*x^7 + 16*x^8 + ...
G.f. = q + 2*q^3 + 5*q^5 + 6*q^7 + 7*q^9 + 12*q^11 + 12*q^13 + 10*q^15 + 16*q^17 + ...

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. A053723, A110712, A129303, A138483, A138512, A138557, A328717.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ m/d KroneckerSymbol[ 5, d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 14 2014 *)

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

Expansion of ( phi(x^5) * psi(x^2) + x * phi(x) * psi(x^10) ) * f(-x^5) * phi(-x^5) / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = (p^(e+1)  + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).
a(n) = A053723(2*n) = A110712(2*n + 1) = A129303(2*n + 1) = A138483(2*n + 1) = A138512(2*n + 1) = A138557(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (5/2) * A328717 = 2*Pi^2/(5*sqrt(5)) = 1.7655285081... . - Amiram Eldar, Nov 23 2023

A138558 Expansion of eta(q^2)^7 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^3) in powers of q.

Original entry on oeis.org

1, 2, -2, -4, 1, -4, -6, 8, 7, 2, 12, 8, -12, -12, -2, -16, -16, 14, 20, -4, 12, 24, -22, -16, 1, -24, -20, 24, 30, -4, 32, 32, -24, -32, -6, -28, -36, 40, 24, 8, 42, 24, -42, -48, 7, -44, -46, 32, 43, 2, 32, 48, -52, -40, 12, -48, -40, 60, 60, 8, 62, 64, -42

Offset: 1

Michael Somos, Mar 24 2008, Mar 25 2008

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

			G.f. = q + 2*q^2 - 2*q^3 - 4*q^4 + q^5 - 4*q^6 - 6*q^7 + 8*q^8 + 7*q^9 + 2*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. 343, Eq. (3.44).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A111580, A138557.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, # Mod[n/#, 2] KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^5 / QPochhammer[ -q^5] - QPochhammer[ q^2]^5 / QPochhammer[ q^10]) / 5, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q]^3 Pochhammer[ q, -q] QPochhammer[ -q^5] QPochhammer[ q^5, -q^5]^3, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};

Expansion of q * (f(q)^3 / chi(q)) * (f(q^5) / chi(q^5)^3) in powers of q where chi(), f() are Ramanujan theta functions.
Expansion of ( f(q)^5 / f(q^5) - f(-q^2)^5 / f(-q^10) ) / 5 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 20 sequence [ 2, -5, 2, -3, 0, -5, 2, -3, 2, -4, 2, -3, 2, -5, 0, -3, 2, -5, 2, -4, ...].
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(5^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138557.
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k))^3 * (1 - x^(5*k)) * (1 - x^(10*k-5)) * (1 + x^(10*k))^2.
G.f.: Sum_{k>0} (-1)^k * ( f(5*k-1) + f(5*k-2) - f(5*k-3) - f(5*k-4) ) where f(k) := k * x^k / (1 - x^(2*k)).
a(n) = -(-1)^n * A111580(n). a(2*n) = 2 * A111580(n). - Michael Somos, Sep 08 2015
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - Amiram Eldar, Feb 01 2024

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A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function.

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A138558 Expansion of eta(q^2)^7 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^3) in powers of q.

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