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 1-3 of 3 results.

A098098 a(n) = sigma(6*n+5)/6.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72
Offset: 0

Views

Author

Vladeta Jovovic, Sep 14 2004

Keywords

Comments

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

Examples

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...

Links

Crossrefs

Cf. A000203, A016969, A033686, A097723, A121443, A125514, A133739, A134077, A134078, A134079, A186100, A232343, A232356, A252650, A238831.

Programs

  • Magma
    Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */
  • Mathematica
    Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)
  • PARI
    a(n) = sigma(6*n + 5)/6
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */

Formula

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) = A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A131946 Expansion of (phi(-q) * phi(-q^3))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, -4, 20, -24, 4, -32, 52, -4, 24, -48, 20, -56, 32, -24, 116, -72, 4, -80, 120, -32, 48, -96, 52, -124, 56, -4, 160, -120, 24, -128, 244, -48, 72, -192, 20, -152, 80, -56, 312, -168, 32, -176, 240, -24, 96, -192, 116, -228, 124, -72, 280, -216, 4
Offset: 0

Views

Author

Michael Somos, Jul 30 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 4*q + 4*q^2 - 4*q^3 + 20*q^4 - 24*q^5 + 4*q^6 - 32*q^7 + 52*q^8 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

Links

Crossrefs

Cf. A034896, A111932, A125514, A131947.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 55); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Nov 11 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ d {0, 1, -1, 0, -1, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Sum[ n/d {6, 1, -3, -2, -3, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Sep 19 2013 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2, n))};
  • PARI
    {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * [0, 1, -1, 0, -1, 1][d%6 + 1]))}; /* Michael Somos, Sep 19 2013 */
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=55) . basis();  A[0] - 4*A[1] + 4*A[2]; # Michael Somos, Sep 19 2013

Formula

Expansion of (4*a(q^2)^2 - a(q)^2) / 3 in powers of q where a() is a cubic AGM theta function.
Expansion of (b(q)^2 / b(q^2)) * (c(q)^2 / c(q^2)) / 3 in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q) * eta(q^3))^4 / ( eta(q^2) * eta(q^6))^2 in powers of q.
Euler transform of period 6 sequence [-4, -2, -8, -2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2^e) = 3 - 2^(e+1), b(3^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Sep 19 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 48 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111932. - Michael Somos, Sep 19 2013
G.f.: 1 - 4 * (Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k)) = (theta_3(-x) * theta_3(-x^3))^2.
a(n) = (-1)^n * A034896(n). a(n) = -4 * A131947(n) unless n = 0.
a(3*n) = a(n). a(2*n) = A125514(n). - Michael Somos, Sep 19 2013

A212817 Theta series of direct sum of 2 copies of 4-dimensional lattice QQF.4.i.

Original entry on oeis.org

1, 8, 56, 168, 536, 624, 2328, 1600, 4184, 4872, 7824, 6432, 19320, 10672, 21568, 22320, 33752, 23184, 62904, 32992, 66000, 61248, 83040, 58944, 155832, 75320, 136912, 130728, 179776, 117168, 291024, 142720, 269528, 236448, 307440, 207744, 528024, 243952
Offset: 0

Views

Author

Michael Somos, May 27 2012

Keywords

Examples

			G.f. = 1 + 8*x + 56*x^2 + 168*x^3 + 536*x^4 + 624*x^5 + 2328*x^6 + 1600*x^7 + ...

Links

Crossrefs

Cf. A028977, A030209, A125514.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 4), 38); A[1] + 8*A[2] + 56*A[3] + 168*A[4] + 536*A[5]; /* Michael Somos, Jun 04 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^6], e2 = QPochhammer[ x^2] QPochhammer[ x^3]}, (e2^7 / e1^5 - x e1^7 / e2^5)^2 ], {x, 0, n}]; (* Michael Somos, Apr 19 2015 *)
  • PARI
    {a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( (B^7 / A^5 - x * A^7 / B^5)^2, n))};
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ]; polcoeff( (1 + 2 * x * Ser( qfrep( G, n, 1)))^2, n))};

Formula

Expansion of ((eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 576 (t/i)^4 f(t) where q = exp(2 Pi i t).
Convolution square of A125514.
a(n) = A028977(n) + 8 * A030209(n). - Michael Somos, Jun 05 2015
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