A089810 -id:A089810 - cp's OEIS frontend

A080995 Characteristic function of generalized pentagonal numbers A001318.

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

Michael Somos, Feb 27 2003

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

			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 + ...

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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Don Zagier, Elliptic modular forms and their applications, in "The 1-2-3 of modular forms", Springer-Verlag, 2008.
Index entries for characteristic functions.

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

a080995 = a033683 . (+ 1) . (* 24)  -- Reinhard Zumkeller, Nov 14 2015

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 *)

{a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))};

{a(n) = issquare( 24*n + 1)}; /* Michael Somos, Apr 13 2005 */

{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))};

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

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

Eric W. Weisstein, Nov 12 2003

sign

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

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.

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 *);

{a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */

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

A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 0, -2, -2, 0, 0, 1, -4, -4, 0, 0, 4, 0, 0, -2, -2, 4, 0, 2, 0, 0, 0, 0, 6, 2, 0, 0, -2, 0, 0, 1, 0, -4, 0, 4, 4, 0, 0, -4, -2, 0, 0, 0, -8, 0, 0, 0, 2, 3, 0, -4, -2, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -4, 0, 0, 2, 0, 0, 0, 4, 4, 2, 0, 0, 0, 0, 0, 2

Offset: 0

Michael Somos, May 25 2015

sign

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

			G.f. = 1 + 2*q + q^2 - 2*q^4 - 2*q^5 + q^8 - 4*q^9 - 4*q^10 + 4*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A004018, A089810, A121444, A258277, A258278.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q]^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^2, n))};

Expansion of eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^2) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 36 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A002175.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^k + x^(2*k))^2.
Convolution square of A089810.
a(2*n) = A258228(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0. a(6*n + 2) = A122865(n). a(6*n + 4) = -2 * A122856(n). a(12*n + 1) = 2 * A002175(n). a(12*n + 5) = -2 * A121444(n).
a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

A204843 Expansion of x * f(-x^24) + (3 * phi(-x^36) - phi(-x^4)) / 2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 19 2012

sign

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

			1 + x + x^4 - x^16 - x^25 - 2*x^36 - x^49 - x^64 + x^100 + x^121 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A089810, A010815, A204853.

a[n_]:= SeriesCoefficient[(3*EllipticTheta[3, 0, -q^36] -EllipticTheta[3, 0, -q^4])/2 + q*QPochhammer[q^24, q^72]*QPochhammer[q^48, q^72]* QPochhammer[q^72, q^72], {q, 0, n}]; Table[a[n], {n,0,100}] (* G. C. Greubel, Dec 19 2017 *)

{a(n) = local(m); if( n<1, n==0, if( issquare( n, &m), (-1)^(m\6) * [ 2, 1, 1, 0, -1, -1][m%6 + 1]))}

Expansion of phi(-x^36) + x * f(-x^24) + x^4 * f(-x^12, -x^60) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of f(-x^9, x^9) + x * f(x^3, -x^15) in powers of x where f() is the two variable Ramanujan theta function.
Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(6*n + 3) = a(8*n + 5) = 0. a(4*n) = A089810(n). a(24*n + 1) = A010815(n). a(25*n) = a(49*n) = A204853(n).

A204853 Expansion of (3 * phi(-x^36) - phi(-x^4)) / 2 - x * f(-x^24) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 20 2012

sign

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

			1 - x + x^4 - x^16 + x^25 - 2*x^36 + x^49 - x^64 + x^100 - x^121 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010815, A089810, A204843.

a[n_]:= SeriesCoefficient[(3*EllipticTheta[3, 0, -q^36] -EllipticTheta[3, 0, -q^4])/2 - q*QPochhammer[q^24, q^72]*QPochhammer[q^48, q^72]* QPochhammer[q^72, q^72], {q, 0, n}]; Table[a[n], {n,0,100}] (* G. C. Greubel, Dec 19 2017 *)

{a(n) = local(m); if( n<1, n==0, if( issquare( n, &m), (-1)^(m\6) * [ 2, -1, 1, 0, -1, 1][m%6 + 1]))}

Expansion of phi(-x^36) - x * f(-x^24) + x^4 * f(-x^12, -x^60) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of f(-x^9, x^9) - x * f(-x^3, x^15) in powers of x where f() is the two variable Ramanujan theta function.
Euler transform of period 24 sequence [ -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, ...].
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(6*n + 3) = a(8*n + 5) =  0. a(4*n) = A089810(n). a(24*n + 1) = - A010815(n). a(25*n) = a(49*n) = A204843(n). a(n) = (-1)^n * A204843(n).

cp's OEIS Frontend

A080995 Characteristic function of generalized pentagonal numbers A001318.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A204843 Expansion of x * f(-x^24) + (3 * phi(-x^36) - phi(-x^4)) / 2 in powers of x where phi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A204853 Expansion of (3 * phi(-x^36) - phi(-x^4)) / 2 - x * f(-x^24) in powers of x where phi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula