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)

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

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

A116916 Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.

Original entry on oeis.org

1, 5, -7, 0, 0, -11, 0, 13, 0, 0, 0, 0, 17, 0, 0, -19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 26 2006

Keywords

Comments

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

Examples

			1 + 5*x - 7*x^2 - 11*x^5 + 13*x^7 + 17*x^12 - 19*x^15 - 23*x^22 + 25*x^26 + ...
q + 5*q^25 - 7*q^49 - 11*q^121 + 13*q^169 + 17*q^289 - 19*q^361 +...

Links

Crossrefs

Cf. A010816, A202394.
Cf. A282937, A282941.

Programs

  • Mathematica
    a[0] = 1; a[n_] := SeriesCoefficient[QPochhammer[x + x*O[x]^(3n)]^3 + 3x * QPochhammer[x^9 + O[x]^(3n)]^3, 3n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 06 2015, adapted from PARI *)
a[ n_] := With[ {m = Sqrt[ 24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ 3, m] KroneckerSymbol[ -3, m], 0]]; (* Michael Somos, Apr 27 2018 *)
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( 3, n) * kronecker( -3, n))};
  • PARI
    {a(n) = if( n<1, n==0, n*=3; polcoeff( eta(x + x * O(x^n))^3 + 3 * x * eta(x^9 + x * O(x^n))^3, n))};

Formula

Expansion of f(-x) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
Expansion of f(-x)^3 + 3 * x * f(-x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A202394.
G.f.: Sum_{k in Z} (-1)^k * (6*k + 1) * x^(k * (3*k + 1) / 2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = 5 * a(n).
a(n) = A010816(3*n).

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
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