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.

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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

Views

Author

Michael Somos, May 25 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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.

A258291 Expansion of q^(-1/4) * eta(q) * eta(q^2) * eta(q^6) / eta(q^3) in powers of q.

Original entry on oeis.org

1, -1, -2, 2, -1, 0, 3, -1, 0, 2, -1, -4, 1, -1, 0, 2, -2, 0, 2, 0, -2, 4, -1, 0, 2, -1, 0, 2, -1, -4, 1, -2, 0, 0, -1, 0, 4, -1, -4, 2, 0, 0, 3, -1, 0, 2, -2, 0, 2, -1, 0, 4, 0, 0, 0, -2, -6, 2, -1, 0, 2, -1, 0, 0, -1, -4, 4, -1, 0, 2, -1, 0, 3, -1, 0, 0, -2
Offset: 0

Views

Author

Michael Somos, May 25 2015

Keywords

Examples

			G.f. = 1 - x - 2*x^2 + 2*x^3 - x^4 + 3*x^6 - x^7 + 2*x^9 - x^10 - 4*x^11 + ...
G.f. = q - q^5 - 2*q^9 + 2*q^13 - q^17 + 3*q^25 - q^29 + 2*q^37 - q^41 + ...

Links

Crossrefs

Cf. A002175, A008441, A121444, A258277.

Programs

  • Mathematica
    QP := QPochhammer; CoefficientList[Series[QP[q]*QP[q^2]*QP[q^6]/QP[q^3], {q, 0, 50}], q] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A), n))};

Formula

Euler transform of period 6 sequence [ -1, -2, 0, -2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 9 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258277.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 + x^(3*k)).
a(3*n) = A002175(n). a(3*n + 1) = - A121444(n). a(9*n + 2) = -2 * A008441(n). a(9*n + 5) = a(9*n + 8) = 0.

A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 0, -2, -8, 0, 0, 4, -4, -4, 0, 0, 4, 0, 0, -2, -8, 4, 0, 8, 0, 0, 0, 0, 6, 8, 0, 0, -8, 0, 0, 4, 0, -4, 0, 4, 4, 0, 0, -4, -8, 0, 0, 0, -8, 0, 0, 0, 2, 12, 0, -4, -8, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -16, 0, 0, 8, 0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Michael Somos, Jun 03 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};
  • PARI
    {a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

Formula

Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).

A256279 Expansion of psi(q) * chi(-q^3) * phi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, 0, -4, -2, 0, 0, 2, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, 2, 0, 0, -2, 0, 0, 0, 0, -8, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, -4, -2
Offset: 0

Views

Author

Michael Somos, Jun 02 2015

Keywords

Comments

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

Examples

			G.f. = 1 + q - q^4 - 4*q^9 - 2*q^10 + 2*q^13 - q^16 + 4*q^18 + 3*q^25 + ...

Links

Crossrefs

Cf. A002175, A122856, A122865, A256269, A258277.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)) QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^9], {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, (-1)^(n\3) * (n%3<2) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
  • PARI
    {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^9 + A)^2 / (eta(x + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^9)^2 / (eta(q) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, -2, -1, 1, -1, 1, -1, 0, -1, 1, -2, ...].
a(n) = (-1)^n * A256269(n). a(4*n) = A256269(n).
a(3*n + 2) = a(4*n + 3) = 0. a(3*n + 1) = A258277(n). a(6*n + 4) = - A122856(n). a(12*n + 1) = A002175(n). a(12*n + 4) = - A122865(n).

A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 3, 6, 7, 6, 6, 7, 12, 12, 12, 9, 6, 12, 12, 18, 13, 12, 18, 12, 18, 12, 13, 18, 12, 24, 12, 12, 24, 21, 30, 12, 18, 18, 12, 24, 18, 19, 18, 24, 24, 18, 24, 36, 24, 18, 19, 18, 24, 24, 30, 18, 12, 36, 30, 24, 21, 18, 36, 24, 36, 24, 12, 36, 36, 36, 18, 25, 30, 24, 24, 24, 30, 24, 36, 30, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 14 2017

Keywords

Comments

Conjecture: every number is the sum of at most k - 4 generalized k-gonal numbers (for k >= 8).
In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m-2)^3 can be written as the sum of five m-gonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper Zhi-Wei Sun proved that each natural number is the sum of four octagonal numbers. - Zhi-Wei Sun, Oct 03 2020

Examples

			a(6) = 7 because we have [5, 1, 0], [5, 0, 1], [2, 2, 2], [1, 5, 0], [1, 0, 5], [0, 5, 1] and [0, 1, 5].

Links

Crossrefs

Cf. A001318, A002175, A008443, A080995, A093518, A093519, A255350, A255934, A256132, A256171, A280718.

Programs

  • Maple
    N:= 100;
bds:= [fsolve(k*(3*k-1)/2 = N)];
G:= add(x^(k*(3*k-1)/2),k=floor(min(bds))..ceil(max(bds)))^3:
seq(coeff(G,x,n),n=0..N); # Robert Israel, Aug 16 2017
  • Mathematica
    nmax = 75; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, -nmax, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Sum[x^((6 k^2 + 6 k + (-1)^(k + 1) (2 k + 1) + 1)/16), {k, 0, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[EllipticTheta[4, 0, x^3]^3/QPochhammer[x, x^2]^3, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.
G.f.: (Sum_{k>=0} x^A001318(k))^3.
G.f.: Product_{n >= 1} ( (1 - q^(3*n))/(1 - q^n + q^(2*n)) )^3. - Peter Bala, Jan 04 2025
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