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 1-9 of 9 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

A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

Original entry on oeis.org

1, -4, 2, 9, -9, -2, 0, -5, 9, 9, 0, -9, -1, -9, 0, -1, 9, 9, -9, 9, 0, 9, -5, -18, -18, 9, 7, 0, 9, 0, 0, 9, 9, -18, 18, -7, -9, -9, -9, 9, -4, -9, -9, 18, 9, 0, 18, 9, 0, -9, -9, -8, -9, 18, -9, 9, -18, 1, -9, -18, 9, 0, 18, 18, 0, 0, 9, -9, 18, -9, 5, -9, 0, -9, -9, -9, -18, 11, 9
Offset: 0

Views

Author

Paul D. Hanna, Feb 16 2011

Keywords

Examples

			G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018

Links

Crossrefs

Cf. A182819, A144613, A000203.
Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 12 2018 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,-sigma(3*m)*x^m/m)+x*O(x^n)),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}

Formula

G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by Joerg Arndt in A182819]
a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017
Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - Michael Somos, Jul 12 2018

A283077 Expansion of Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22527, 62329, 164516, 416948, 1019690, 2416246, 5565864, 12498215, 27421815, 58903768, 124088548, 256749822, 522450250, 1046735092, 2066948472, 4026431543, 7743987036, 14715788745, 27648250012, 51390298666, 94550761844
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2017

Keywords

Examples

			G.f.: A(x) = 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + ...
log(A(x)) = 8*x + 24*x^2/2 + 32*x^3/3 + 56*x^4/4 + 48*x^5/5 + 96*x^6/6 + 57*x^7/7 + 120*x^8/8 + ... + sigma(7*n)*x^n/n + ...

Links

Crossrefs

Cf. A282942 (Product_{n>=1} (1 - x^n)^8/(1 - x^(7*n))), A283078 (sigma(7*n)).
Cf. exp( Sum_{n>=1} sigma(k*n)*x^n/n ): A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), A283119 (k=6), this sequence (k=7), A283120 (k=8), A283121 (k=9).

Formula

G.f.: exp( Sum_{n>=1} sigma(7*n)*x^n/n ).
a(n) = (1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 3025 * exp(sqrt(110*n/21)*Pi) / (28224*sqrt(14)*n^(5/2)). - Vaclav Kotesovec, Mar 20 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

A282941 a(n) = A000730(7*n).

Original entry on oeis.org

1, 41, -176, 98, 322, -181, -140, -489, 112, 889, 14, -560, 125, 154, 756, -1317, -1778, 1554, -1218, 2688, 1764, -980, 71, -1575, 14, -1638, -419, 56, -1988, -2716, 6223, 6860, 1302, -700, -3416, -4733, -2548, -4725, 3836, 1106, 2631, 5096, -5656, 2660, -7875
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2017

Keywords

Examples

			G.f.: 1 + 41*q - 176*q^2 + 98*q^3 + 322*q^4 - 181*q^5 - 140*q^6 - 489*q^7 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 191.

Links

Crossrefs

Cf. A000730, A116916, A282937, A282942.

Formula

G.f.: Product_{n>=1} (1 - q^n)^8/(1 - q^(7*n)) + 49*q*(Product_{n>=1} (1 - q^n)^4*(1 - q^(7*n))^3).
a(n) = (-1)^j mod 7 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 7.
a(n) = A282942(n) mod 49.

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
Showing 1-9 of 9 results.

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