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 1-10 of 10 results.

A138532 Expansion of psi(x) / psi(x^5) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 2, 0, 0, 1, 0, -2, -1, 0, -2, 0, 3, 2, 0, 3, 0, -5, -2, 0, -3, 0, 6, 2, 0, 4, 0, -8, -3, 0, -6, 0, 11, 5, 0, 8, 0, -14, -6, 0, -10, 0, 18, 6, 0, 12, 0, -22, -9, 0, -16, 0, 28, 13, 0, 21, 0, -36, -14, 0, -25, 0, 44, 16, 0, 30, 0
Offset: 0

Views

Author

Michael Somos, Mar 23 2008

Keywords

Comments

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

Examples

			G.f. = 1 + x + x^3 - x^5 - x^8 + 2*x^10 + x^13 - 2*x^15 - x^16 - 2*x^18 + ...
G.f. = 1/q + q + q^5 - q^9 - q^15 + 2*q^19 + q^25 - 2*q^29 - q^31 - 2*q^35 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 233, Entry 66.

Links

Crossrefs

Cf. A116494, A138516.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, 0, x^(1/2)] / EllipticTheta[ 2, 0, x^(5/2)], {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) / eta(x + A) * (eta(x^2 + A) / eta(x^10 + A))^2, n))};

Formula

Expansion of q^(1/2) * (eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q) in powers of q.
Euler transform of period 10 sequence [ 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - u^2)^2 - (u^2 - 1) * (u^2 - 5) * v^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (u^2 - 1) * (v^2 - 5).
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * w * (v^2 - 1) - v * (v + w)^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u2 * u6 * (u3^2 - u1^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138526.
G.f.: (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-1) where P(n,x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 4) = 0.
Convolution square is A138516. Convolution inverse is A116494.

A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 2, 2, -4, 0, 0, 2, 4, -8, 0, 0, 4, 8, -14, 0, 0, 8, 14, -24, 0, 0, 12, 22, -40, 0, 0, 20, 36, -64, 0, 0, 32, 56, -98, 0, 0, 48, 84, -148, 0, 0, 72, 126, -220, 0, 0, 106, 184, -320, 0, 0, 152, 264, -460, 0, 0, 216, 376, -652, 0, 0, 306, 528, -912, 0, 0, 424, 732, -1264, 0, 0, 584, 1008
Offset: 0

Views

Author

Michael Somos, Mar 23 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016

Examples

			G.f. = 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.

Links

Crossrefs

Cf. A116494, A138158, A138526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))};

Formula

Expansion of (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.
Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116494.
G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse is A138526. Convolution square is A138518.

A138517 Expansion of (phi(-q^5) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 12, 32, 76, 164, 336, 656, 1228, 2228, 3932, 6768, 11408, 18872, 30688, 49152, 77644, 121096, 186684, 284720, 429916, 643168, 953904, 1403312, 2048784, 2969764, 4275656, 6116480, 8696864, 12294680, 17285776, 24176288, 33645132
Offset: 0

Views

Author

Michael Somos, Mar 23 2008

Keywords

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			1 + 4*q + 12*q^2 + 32*q^3 + 76*q^4 + 164*q^5 + 336*q^6 + 656*q^7 + ...

Links

Crossrefs

Cf. 4 * A095846(n) = a(n) unless n=0. Convolution inverse of A138518. Convolution square of A138526.

Programs

  • Mathematica
    eta[x_] := x^(1/24)*QPochhammer[x]; A138517[n_] := SeriesCoefficient[ ((eta[q^5]/eta[q])^2*eta[q^2]/eta[q^10])^2, {q, 0, n}]; Table[ A138517[n], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A))^2, n))}

Formula

Expansion of ( (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) )^2 in powers of q.
Euler transform of period 10 sequence [ 4, 2, 4, 2, 0, 2, 4, 2, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u - 1) - 4 * u * v * (v - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138516.
G.f.: (Product_{k>0} P(5, x^k) / P(10, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 3/10 + (1/10)*sqrt(5) + (1/10)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 04 2021

A146162 Expansion of eta(q^2)^2 * eta(q^5) / (eta(q) * eta(q^4)^2) in powers of q.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 0, 2, 4, 3, 0, 3, 8, 4, 0, 6, 14, 8, 0, 10, 22, 12, 0, 16, 36, 21, 0, 25, 56, 30, 0, 38, 84, 48, 0, 57, 126, 68, 0, 84, 184, 102, 0, 121, 264, 143, 0, 172, 376, 207, 0, 243, 528, 284, 0, 338, 732, 400, 0, 465, 1008, 542, 0, 636, 1374, 744, 0, 862, 1856, 996, 0
Offset: 0

Views

Author

Michael Somos, Oct 27 2008

Keywords

Examples

			1 + q + q^3 + 2*q^4 + q^5 + 2*q^7 + 4*q^8 + 3*q^9 + 3*q^11 + 8*q^12 + ...

Links

Crossrefs

A138526(n) = a(4*n). A145722(n) = a(4*n + 1). A146163(n) = a(4*n + 3).

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[QPochhammer[x^5]/(QPochhammer[x]* QPochhammer[ -x^2, x^2]^2), {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A)^2), n))}

Formula

Euler transform of period 20 sequence [ 1, -1, 1, 1, 0, -1, 1, 1, 1, -2, 1, 1, 1, -1, 0, 1, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (4/5)^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A146164.
a(4*n + 2) = 0.

A261796 Expansion of Product_{k>=1} (1+x^k)/((1+x^(3*k))*(1+x^(5*k))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 4, 4, 5, 6, 7, 8, 9, 9, 10, 12, 13, 14, 15, 16, 17, 20, 23, 24, 26, 28, 30, 33, 37, 40, 42, 46, 50, 55, 60, 65, 68, 72, 79, 86, 93, 101, 108, 114, 123, 134, 144, 153, 164, 174, 186, 203, 219, 233, 247, 263, 280
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 01 2015

Keywords

Links

Crossrefs

Cf. A103257, A098151, A138526, A261797.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Product[(1+x^k)/((1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(7*n/5)/3) * 7^(1/4) / (2*sqrt(3) * 5^(1/4) * n^(3/4)).

A261797 Expansion of Product_{k>=1} (1-x^(3*k))*(1-x^(5*k))/(1-x^k).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 7, 11, 12, 16, 19, 25, 29, 37, 43, 55, 63, 78, 90, 110, 127, 153, 176, 211, 242, 286, 328, 386, 441, 515, 586, 682, 775, 895, 1016, 1169, 1323, 1514, 1711, 1953, 2201, 2502, 2815, 3191, 3582, 4048, 4536, 5113, 5719, 6429, 7179, 8052
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 01 2015

Keywords

Links

Crossrefs

Cf. A103257, A098151, A138526, A261796.

Programs

  • Mathematica
    nmax=80; CoefficientList[Series[Product[(1-x^(3*k))*(1-x^(5*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(14*n/5)/3) / sqrt(30*n).

A261968 Expansion of phi(q^5) / phi(q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -8, 14, -22, 36, -56, 84, -126, 184, -264, 376, -528, 732, -1008, 1374, -1856, 2492, -3320, 4394, -5784, 7568, -9848, 12756, -16442, 21096, -26960, 34312, -43500, 54956, -69184, 86804, -108576, 135392, -168336, 208722, -258096, 318320, -391632
Offset: 0

Views

Author

Michael Somos, Sep 06 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 - 8*q^3 + 14*q^4 - 22*q^5 + 36*q^6 - 56*q^7 + ...

Links

Crossrefs

Cf. A138526, A144377.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5 / (eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^5 / (eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2) in powers of q.
Euler transform of period 20 sequence [ -2, 3, -2, 1, 0, 3, -2, 1, -2, 0, -2, 1, -2, 3, 0, 1, -2, 3, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - 2*u^2 + 5*u^4) * (1 - 2*v^2 + 5*v^4) - 4*(u^2 + 2*u*v - v^2)^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 + 3*u*v - u^2) * (u^2 + v^2) - u*v * (1 + 5*u^2*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144377.
G.f.: Product_{k>0} P(10, x^k)^3 * P(5, x^k) / P(20, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.
a(n) = (-1)^n * A138526(n). Convolution inverse is A144377.

A147702 Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, 2, -1, 0, -2, 4, -3, 0, -3, 8, -4, 0, -6, 14, -8, 0, -10, 22, -12, 0, -16, 36, -21, 0, -25, 56, -30, 0, -38, 84, -48, 0, -57, 126, -68, 0, -84, 184, -102, 0, -121, 264, -143, 0, -172, 376, -207, 0, -243, 528, -284, 0, -338, 732, -400, 0, -465, 1008, -542, 0, -636, 1374, -744, 0, -862, 1856, -996, 0
Offset: 0

Views

Author

Michael Somos, Nov 10 2008

Keywords

Comments

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

Examples

			G.f. = 1 - q - q^3 + 2*q^4 - q^5 - 2*q^7 + 4*q^8 - 3*q^9 - 3*q^11 + 8*q^12 + ...

Links

Crossrefs

Cf. A036026, A138526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -q^5] QPochhammer[ q, q^2] / QPochhammer[ q^4], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of chi(-q) * f(q^5) / f(-q^4) in powers of q where f(), chi() are Ramanujan theta functions.
Euler transform of period 20 sequence [ -1, 0, -1, 1, 0, 0, -1, 1, -1, -2, -1, 1, -1, 0, 0, 1, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147699.
a(2*n + 1) = - A036026(n). a(4*n) = A138526(n). a(4*n + 2) = 0.

A262050 Expansion of f(-x)^2 * f(-x^10) / phi(-x)^3 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 11, 28, 63, 132, 264, 504, 928, 1660, 2892, 4924, 8221, 13480, 21750, 34592, 54288, 84168, 129048, 195816, 294282, 438324, 647413, 948748, 1380107, 1993632, 2860984, 4080172, 5784560, 8154900, 11435142, 15953124, 22147824, 30604868, 42102636, 57672312
Offset: 0

Views

Author

Michael Somos, Sep 09 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 11*x^2 + 28*x^3 + 63*x^4 + 132*x^5 + 264*x^6 + 504*x^7 + ...
G.f. = q + 4*q^3 + 11*q^5 + 28*q^7 + 63*q^9 + 132*q^11 + 264*q^13 + ...

Links

Crossrefs

Cf. A138526, A261968.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^10] / EllipticTheta[ 4, 0, x]^3, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^10 + A) / eta(x + A)^4, n))};

Formula

Expansion of q^(-1/2) * eta(q^2)^3 * eta(q^10) / eta(q)^4 in powers of q.
Euler transform of period 10 sequence [ 4, 1, 4, 1, 4, 1, 4, 1, 4, 0, ...].
2 * a(n) = A138526(2*n + 1) = - A261968(2*n + 1).

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)*(1 - x^(n*k))/((1 - x^k)*(1 + x^(n*k))).

Original entry on oeis.org

1, 0, 2, 6, 12, 22, 38, 62, 98, 152, 230, 342, 502, 726, 1038, 1470, 2060, 2862, 3946, 5398, 7334, 9902, 13286, 17726, 23526, 31064, 40822, 53406, 69566, 90246, 116622, 150142, 192610, 246254, 313806, 398638, 504884, 637590, 802934, 1008446, 1263270, 1578526, 1967694, 2447062
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Links

Crossrefs

Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]
Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]
Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]
nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x) - 2*x/(1 - x), where theta_4() is the Jacobi theta function.
a(n) ~ exp(Pi*sqrt(n)) / (8*n). - Vaclav Kotesovec, May 19 2018
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