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 1-4 of 4 results.

A080995 Characteristic function of generalized pentagonal numbers A001318.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 27 2003

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A090864(n))=0. - Reinhard Zumkeller, Apr 22 2006
Starting with offset 1 with 1's signed (++--++,...), i.e., (1, 1, 0, 0, -1, 0, -1, 0, ...); is the INVERTi transform of A000041 starting (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 17 2013
Number 9 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + ...
G.f. = q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...

References

  • Percy A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 81, Article 331.

Links

Crossrefs

Cf. A001318 (support), A010815 (absolute values), A033683, A089806.
Cf. A000326, A180447, A085141, A133985.
Cf. A000009, A000041, A000122, A000700, A010054, A090864, A121373.

Programs

  • Haskell
    a080995 = a033683 . (+ 1) . (* 24)  -- Reinhard Zumkeller, Nov 14 2015
  • Mathematica
    a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> x^(1/2)}, {x, 0, n}]]; (* Michael Somos, Nov 18 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 08 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ IntegerQ[ Sqrt[ 24 n + 1]]]]; (* Michael Somos, Jun 08 2013 *)
  • PARI
    {a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))};
  • PARI
    {a(n) = issquare( 24*n + 1)}; /* Michael Somos, Apr 13 2005 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};

Formula

Expansion of phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 14 2007
Expansion of psi(x) - x * psi(x^9) in powers of x^3 where psi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.
a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p>3. - Michael Somos, Jun 06 2005
Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089810.
G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos, Jan 26 2008
G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].
a(n) = |A010815(n)| = A089806(2*n) = A033683(24*n + 1).
For n > 0, a(n) = b(n) - b(n-1) + c(n) - c(n-1), where b(n) = floor(sqrt(2n/3+1/36)+1/6) (= A180447(n)) and c(n) = floor(sqrt(2n/3+1/36)-1/6) (= A085141(n)). - Mikael Aaltonen, Mar 08 2015
a(n) = (-1)^n * A133985(n). - Michael Somos, Jul 12 2015
a(n) = A000009(n) (mod 2). - John M. Campbell, Jun 29 2016
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2*sqrt(2/3) = 1.632993... . - Amiram Eldar, Jan 13 2024

Extensions

Minor edits by N. J. A. Sloane, Feb 03 2012

A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1
Offset: 0

Views

Author

Eric W. Weisstein, Nov 12 2003

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^5, b = x. - Michael Somos, Jul 12 2012
Number 11 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...

Links

Crossrefs

Related to the 14 primitive eta-products which are holomorphic modular forms of weight 1/2: A000122, A002448, A010054, A010815, A080995, A089801, A089802, this sequence, A089810, A089812, A106459, A121373, A133985, A133988. - Seiichi Manyama, May 15 2017
Cf. A000700, A000122, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, 0, Sqrt[ -q]] / (2 (-q)^(1/8)), {q, 0, n}] (* Michael Somos, Jul 12 2012 *);
  • PARI
    {a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */

Formula

a(n) = -b(n) where b() is multiplicative with b(3^e) = -2(1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...].
G.f.: (Sum_{k in Z} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))). (End)
Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26 2006
Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007
Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089801.
a(n) = (-1)^n * A089810(n). - Michael Somos, Jan 20 2012
For n > 0, a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(2-4*sin(floor(sqrt(n))*Pi/3)^2). - Mikael Aaltonen, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

A133988 Expansion of phi(x) / chi(x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, -1, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 01 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^2, b = -x. - Michael Somos, Jan 21 2012
Number 13 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 + 2*x - x^3 + x^6 - 2*x^10 - x^15 - x^21 - 2*x^28 + x^36 - x^45 + ...
G.f. = q + 2*q^9 - q^25 + q^49 - 2*q^81 - q^121 - q^169 - 2*q^225 + q^289 - ...

Links

Crossrefs

Cf. A089812, A133985.

Programs

  • Mathematica
    a[ n_] := (-1)^Quotient[n, 3] Mod[n + 1, 3] Boole[ IntegerQ[ Sqrt[8 n + 1]]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]  QPochhammer[ x^3, -x^3], {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = (-1)^(n\3) * ((n + 1)%3) * issquare( 8*n + 1)};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Formula

Expansion of q^(-1/8) * eta(q^2)^5 * eta(q^3) * eta(q^12) / ( eta(q) * eta(q^4) * eta(q^6) )^2 in powers of q.
Expansion of psi(-x) + 3 * x * psi(-x^9) in powers of x where psi() is a Ramanujan theta function.
Expansion of f(x, x^5) * f(x) / f(-x^6) = f(-x^3, x^6) + x * f(1, -x^9) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of phi(x^9) / chi(x^3) + 2 * x * psi(-x^9) in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 2, -3, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, ...].
a(n) = b(8*n + 1) where b() is multiplicative with b(3^(2*e)) = -2 * (-1)^e, b(p^(2*e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2*e)) = +1 if p == 1, 7 (mod 8) and b(p^(2*e-1)) = b(2^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133985.
G.f.: Sum_{k>0} (-1)^floor(k/2) * (x^((k^2 - k)/2) + 3 * x^(9*(k^2 - k)/2 + 1) ).
G.f.: Sum_{k in Z} (-1)^(k + floor(k/2)) * x^(3*k * (3*k + 1) / 2) * ( x^(-3*k) + x^(3*k + 1) ).
a(n) = (-1)^n * A089812(n).
a(3*n) = A133985(n). a(3*n + 2) = 0. - Michael Somos, Oct 30 2015

A187615 Expansion of f(-x^17, -x^19) + x^4 * f(-x, -x^35) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Mar 11 2011

Keywords

Comments

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

Examples

			G.f. = 1 + x^4 - x^5 - x^17 - x^19 - x^39 + x^42 + x^70 + x^74 + x^110 + ...
G.f. = q + q^289 - q^361 - q^1225 - q^1369 - q^2809 + q^3025 + q^5041 + q^5329 + ...

Links

Crossrefs

Cf. A133985.

Programs

  • Mathematica
    QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[q^36, q^36]*(QP[q^17, q^36]*QP[q^19, q^36] + q^4*QP[q, q^36]*QP[q^35, q^36]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 04 2018 *)
  • PARI
    {a(n) = my(m); if( n<0, 0, if( issquare( 72*n + 1, &m), if( m%36 != 1, m=-m); if( m%36 == 1, m = (m-1) / 36, if( m%36 != 19, m=-m); if( m%36! = 19, return(0));  m = (m+17) / 36); (-1)^m))};
  • PARI
    {a(n) = my(A); if( n<0, 0, n*=3; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^2, n))};

Formula

Euler transform of period 36 sequence [ 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -2, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, ...].
G.f.: Sum_{k in Z} (-1)^k * x^(18*k^2) * (x^(17*k + 4) + x^k).
a(n) = A133985(3*n). a(5*n + 1) = a(5*n + 3) = 0.
Showing 1-4 of 4 results.

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