cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 07 2007

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

Cf. A053723, A110712, A129303, A138483, A138512, A138557, A328717.
Cf. A000122, A000700, A010054, A121373.

Programs

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

Formula

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

A138557 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^7 / (eta(q^2)^3 * eta(q^5)^2 * eta(q^20)^2) in powers of q.

Original entry on oeis.org

1, -2, 2, -4, 5, -4, 6, -8, 7, -10, 12, -8, 12, -12, 10, -16, 16, -14, 20, -20, 12, -24, 22, -16, 25, -24, 20, -24, 30, -20, 32, -32, 24, -32, 30, -28, 36, -40, 24, -40, 42, -24, 42, -48, 35, -44, 46, -32, 43, -50, 32, -48, 52, -40, 60, -48, 40, -60, 60, -40
Offset: 1

Views

Author

Michael Somos, Mar 24 2008

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A129303, 138558.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5]^5 / QPochhammer[ -q] - q^2 QPochhammer[ q^10]^5 / QPochhammer[ q^2], {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q, -q]^3 QPochhammer[ -q^5] ^3 QPochhammer[ q^5, -q^5], {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
  • PARI
    {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, n/d * kronecker(20, d)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^7 / (eta(x^2 + A)^3 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};

Formula

Expansion of q * (f(q) / chi(q)^3) * (f(q^5)^3 / chi(q^5)) in powers of q where chi(), f() are Ramanujan theta functions.
Expansion of q * f(q^5)^5 / f(q) - q^2 * f(-q^10)^5 / f(-q^2) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 20 sequence [ -2, 1, -2, -1, 0, 1, -2, -1, -2, -4, -2, -1, -2, 1, 0, -1, -2, 1, -2, -4, ...].
a(n) is multiplicative with a(2^e) = -2^e 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 A138558.
G.f.: Sum_{k>0} -(-1)^k * k * x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(10*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(2*k-1)) * (1 + x^(2*k))^2 * (1 + x^(10*k-5))^2 * (1 - x^(10*k))^3.
G.f.: Sum_{k>0} f(10*k-1) - f(10*k-3) - f(10*k-7) + f(10*k-9) where f(k) := x^k / (1 + x^k)^2.
a(n) = -(-1)^n * A129303(n).

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

Views

Author

Michael Somos, Aug 08 2007, Mar 20 2008

Keywords

Comments

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

Examples

			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 +...

References

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

Links

Crossrefs

Cf. A112274, A113185, A129303.
Cf. A000122, A000700, A010054, A121373.

Programs

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

Formula

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
Showing 1-3 of 3 results.

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