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 1-8 of 8 results.

A115110 Expansion of q^(-1/24) * eta(q)^3 / eta(q^2) in powers of q.

Original entry on oeis.org

1, -3, 1, 2, 2, -1, -4, 1, -2, 0, 2, 4, -1, 2, -2, -1, 0, -2, -2, -2, 0, 4, 1, 0, 2, -2, 5, 0, -2, 0, 0, -4, -2, 0, 0, -3, 4, 0, 0, -2, 1, 4, 2, 2, 0, 0, 0, -2, -2, 0, 2, -3, -2, 0, -2, 2, -4, 1, 0, 0, 0, 4, 2, 0, 4, 0, -4, 2, 0, 2, -1, 0, 0, 2, -2, -2, -6, -1, 2, 0, 0, -4, 0, 2, 2, 0, 0, 2, -2, 2, 2, 0, 1, 0, 0, 2, 4, 0, 0, -2, 1, -6, 0, -2, 0
Offset: 0

Views

Author

Michael Somos, Mar 07 2006

Keywords

Comments

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

Examples

			G.f. = 1 - 3*x + x^2 + 2*x^3 + 2*x^4 - x^5 - 4*x^6 + x^7 - 2*x^8 + 2*x^10 + ...
G.f. = q - 3*q^25 + q^49 + 2*q^73 + 2*q^97 - q^121 - 4*q^145 + q^169 - 2*q^193 + ...

References

  • B. Gordon and D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989; see page 182. MR1019331 (90k:11050)

Links

Crossrefs

Cf. A107033, A002107.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), this sequence (k=2), A185654 (k=3), A282937 (k=5), A282942 (k=7).

Programs

  • Magma
    m:=120; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^2 / (1 + x^j): j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018
  • Maple
    prod := n -> mul( (1 - x^k)^2*(1 - x^(2*k-1)), k = 1..n):
a := n -> coeff(prod(100), x, n):
seq(a(n), n = 0..100); # Peter Bala, Jan 01 2021
  • Mathematica
    a[ n_] :=  SeriesCoefficient[ QPochhammer[ x]^3 / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] :=  SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x], {x, 0, 2 n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] :=  SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Jul 12 2012 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / eta(x^2 + A), n))};
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(120)
s = prod((1 - x^j)^2 / (1 + x^j) for j in (1..120))
s.coefficients() # G. C. Greubel, Nov 18 2018

Formula

Expansion of f(x) * f(-x) in powers of x^2 where f() is a Ramanujan theta function.
Expansion of f(-x) * phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.
Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A4 + A1*A3 + A5*A6 + 4*A2^2, A2 = x^2 * A(x^49).
Euler transform of period 2 sequence [ -3, -2,...].
G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^k).
G.f.: Sum_{k>=0} ( x^((3*k^2 + k)/2) * (1 - x^(2*k + 1)) * Sum_{|j|<=k} (-x)^(-j^2) ).
a(49*n + 2) = a(n). a(7*n + 2) = 0 unless n = 7*k.
a(n) = (-1)^n * A107033(n).
G.f.: exp( Sum_{n>=1} -sigma(2*n)*x^n/n ). - Seiichi Manyama, Mar 02 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
From Peter Bala, Jan 01 2021: (Start)
For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/24) = e*a(n), where e = 1 if p == 7 or 23 (mod 24) and e = -1 if p == 11 or 19 (mod 24).
If n > 0 and p are coprime then a(n*p + (p^2 - 1)/24) = 0. Cf. A002107.
(End)

A282937 a(n) = A000728(5*n).

Original entry on oeis.org

1, -6, 9, 10, -30, 1, 5, 51, 10, -100, 20, -55, 109, 110, -130, -1, -110, 160, 10, -230, 100, 15, 191, 120, -230, -100, -89, 160, 90, -340, 120, 5, 300, 200, -260, -1, -275, 240, -100, -270, 119, -165, 260, 410, -200, -40, 20, 200, -110, -500, 180, -54, 140
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2017

Keywords

Comments

This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q){infinity}/(q^25; q^25){infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5){infinity}/(q^25; q^25){infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.

Examples

			G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.

Links

Crossrefs

Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j,1,62}], {x,0,60}], x] (* G. C. Greubel, Nov 18 2018 *)
  • PARI
    m=60; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
prec = 60
x = R.gen().O(prec)
s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).
a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.
Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - Seiichi Manyama, Feb 28 2017
G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 1, -133, -140, 308, -70, 174, 56, -518, -141, -63, 868, -140, 238, 294, -1029, -1154, -203, 2366, -658, 1296, 350, -1547, -1295, -1666, 3234, -2128, 2534, 2464, -2577, -3087, -609, 5600, -2716, 2435, 294, -3745, -4249, -1015, 8526
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2017

Keywords

Examples

			G.f.: 1 - 8*q + 20*q^2 - 70*q^4 + 64*q^5 + 56*q^6 + q^7 - 133*q^8 - 140*q^9 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 192.

Links

Crossrefs

Cf. A282941.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), A282937 (k=5), this sequence (k=7).

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // G. C. Greubel, Nov 18 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^k)^8/(1-x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Nov 18 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(j=1,m, (1-x^j)^8/(1-x^(7*j)))) \\ G. C. Greubel, Nov 18 2018
  • Sage
    R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = prod((1-x^j)^8/(1-x^(7*j)) for j in (1..50))
list(s) # G. C. Greubel, Nov 18 2018

Formula

G.f.: exp( Sum_{n>=1} -sigma(7*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283164 Expansion of exp( Sum_{n>=1} -sigma(6*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -12, 58, -133, 95, 194, -418, 97, 325, -99, -238, 169, -217, 131, 190, -145, 441, -647, 169, -527, 72, 1129, 313, -972, 2, -491, -565, 1944, -1175, -216, 972, 863, -1259, 288, 0, -1155, -1355, -207, 2925, 1753, 1402, -2387, -2257, -1030, 315, 432, -72, 1621, 358
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2017

Keywords

Links

Crossrefs

Cf. A224613 (sigma(6*n)), A283119 (exp( Sum_{n>=1} sigma(6*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), this sequence (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^n)^12 * (1 - x^(6*n))/((1 - x^(2*n))^4 * (1 - x^(3*n))^3).
a(n) = -(1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283168 Expansion of exp( Sum_{n>=1} -sigma(8*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -15, 97, -350, 770, -1133, 1540, -2731, 4230, -3960, 3402, -6580, 9167, -5390, 4310, -11061, 12320, -5306, 2030, -7530, 14784, -4340, -10119, -9240, 20090, 11438, -17275, -4928, 2270, 14080, -26840, 7700, 16646, 24640, -53760, 7449, 10780, 46200, -61600
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2017

Keywords

Links

Crossrefs

Cf. A283120 (exp( Sum_{n>=1} sigma(8*n)*x^n/n )), A283122 (sigma(8*n)).
Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), this sequence (k=8), A283169 (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^n)^15/(1 - x^(2*n))^7.
a(n) = -(1/n)*Sum_{k=1..n} sigma(8*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283169 Expansion of exp( Sum_{n>=1} -sigma(9*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -13, 65, -126, -117, 988, -1377, -1157, 5382, -4419, -4212, 12519, -11179, -5058, 27378, -23005, -16488, 44343, -30249, -18513, 73710, -56259, -38741, 93483, -69570, -23778, 137266, -90396, -74079, 140292, -108621, -39249, 222624, -145710, -99234
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2017

Keywords

Links

Crossrefs

Cf. A283121 (exp( Sum_{n>=1} sigma(9*n)*x^n/n )), A283123 (sigma(9*n)).
Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), this sequence (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^n)^13/(1 - x^(3*n))^4.
a(n) = -(1/n)*Sum_{k=1..n} sigma(9*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283163 Expansion of exp( Sum_{n>=1} -sigma(4*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -7, 17, -14, 2, -21, 36, 13, -26, -24, 10, 12, -17, 34, 22, 19, -96, -10, 14, 38, 0, 12, -23, 72, -38, -2, -11, -64, -34, 0, 72, 84, -26, 0, 0, -79, 60, 24, -32, -58, -7, -84, 50, 26, 120, 0, 0, 46, -34, -64, 10, -119, 70, 0, 22, -70, 36, 37, -120, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 02 2017

Keywords

Links

Crossrefs

Cf. A182820 (exp( Sum_{n>=1} sigma(4*n)*x^n/n )), A193553 (sigma(4*n)).
Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), this sequence (k=4), A282937 (k=5), A283164 (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

Formula

G.f.: Product_{n>=1} (1 - x^n)^7/(1 - x^(2*n))^3.
a(n) = -(1/n)*Sum_{k=1..n} sigma(4*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283243 Expansion of exp( Sum_{n>=1} -sigma_2(3*n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -10, 25, 53, -270, -77, 1057, 610, -2031, -5438, -1953, 17236, 34121, 3351, -103369, -195850, -55471, 468448, 1067785, 764094, -1430780, -4974559, -6242563, 334620, 16946199, 34459888, 29243953, -24503978, -124514921, -205795663, -140256312, 191109263
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2017

Keywords

Links

Crossrefs

Cf. A283237 (sigma_2(3*n)), A283238 (exp( Sum_{n>=1} sigma_2(3*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(3*n)*x^n/n ): A185654 (k=1), this sequence (k=2).
Cf. exp( Sum_{n>=1} -sigma_2(m*n)*x^n/n ): A073592 (m=1), A283242 (m=2), this sequence (m=3).

Programs

  • PARI
    A283243_vec(m)=Vec(exp(sum(n=1,m,-sigma(3*n,2)*x^n/n)+x*O(x^m))) \\ Yields m+1 terms a(0..m). - M. F. Hasler, Mar 05 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} sigma_2(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Showing 1-8 of 8 results.

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