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 1-2 of 2 results.

A085261 Expansion of chi(x) / phi(x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -2, -1, 5, 3, -9, -5, 18, 10, -30, -16, 53, 29, -85, -44, 139, 73, -215, -110, 335, 172, -502, -253, 755, 382, -1104, -550, 1614, 805, -2312, -1142, 3305, 1631, -4650, -2277, 6525, 3193, -9041, -4395, 12486, 6063, -17070, -8247, 23255, 11218, -31414, -15090, 42289, 20285
Offset: 0

Views

Author

Michael Somos, Jun 23 2003

Keywords

Comments

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

Examples

			G.f. = 1 + x - 2*x^2 - x^3 + 5*x^4 + 3*x^5 - 9*x^6 - 5*x^7 + 18*x^8 + 10*x^9 - ...
G.f. = 1/q + q^23 - 2*q^47 - q^71 + 5*q^95 + 3*q^119 - 9*q^143 - 5*q^167 + 18*q^191 + ...

Links

Crossrefs

Cf. A000041, A097566, A190101, A246712.

Programs

  • Maple
    t1:=mul( (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))^2), n=1..100): t2:=series(t1,q,100): f:=n->coeff(t2,q,n); # N. J. A. Sloane, Jan 25 2009
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / EllipticTheta[ 3, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 01 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ -x^2]^2), {x, 0, n}]; (* Michael Somos, Sep 02 2014 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / eta(x + A) / eta(x^4 + A)^6, n))};
  • PARI
    {a(n) = polcoeff( prod( k=1,( n+1)\2, 1 + x^(2*k - 1), 1 + x * O(x^n)) / prod(k=1, (n+2)\4, (1 - x^(4*k)) * (1 + x^(4*k - 2))^2, 1 + x * O(x^n)), n)};

Formula

Expansion of psi(x) / f(x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2014
Expansion of q^(1/24) * eta(q^2)^4 * eta(q^8)^2 / (eta(q) * eta(q^4)^6) in powers of q.
Euler transform of period 8 sequence [1, -3, 1, 3, 1, -3, 1, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246712. - Michael Somos, Sep 02 2014
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / ((1 - x^(4*k)) * (1 + x^(4*k - 2))^2).

A097566 Number of partitions p of n for which Odd(p) = Odd(p') (mod 4), where p' is the conjugate of p.

Original entry on oeis.org

1, 1, 0, 1, 5, 5, 1, 5, 20, 20, 6, 20, 65, 65, 25, 66, 185, 185, 85, 190, 481, 482, 250, 501, 1165, 1170, 666, 1230, 2666, 2685, 1646, 2850, 5827, 5887, 3830, 6303, 12251, 12415, 8487, 13395, 24912, 25323, 18052, 27507, 49215, 50176, 37072, 54832, 94781, 96905
Offset: 0

Views

Author

Wouter Meeussen, Aug 28 2004

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Odd(p) is the number of odd parts of a partition p. a(n) is denoted t(n) in Problem 10969.

Examples

			G.f. = 1 + x + x^3 + 5*x^4 + 5*x^5 + x^6 + 5*x^7 + 20*x^8 + 20*x^9 + 6*x^10 + ...
G.f. = 1/q + q^23 + q^71 + 5*q^95 + 5*q^119 + q^143 + 5*q^167 + 20*q^191 + 20*q^215 + ...
a(5) = 5 because only the partitions {5}, {3,2}, {3,1,1}, {2,2,1}, {1,1,1,1,1} have conjugates resp. {1,1,1,1,1}, {2,2,1}, {3,1,1}, {3,2}, {5} with matching counts of odd elements (resp. (1,5), (1,1), (3,3), (1,1), (5,1) being congruent modulo 4 ).

Links

Crossrefs

Cf. A000041, A085261, A097567, A112128, A190101.

Programs

  • Maple
    with(combinat); t1:=mul( (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))^2), n=1..100): t2:=series(t1,q,100): f:=n->coeff(t2,q,n); p:=numbpart; t:=n->(p(n)+f(n))/2; # N. J. A. Sloane, Jan 25 2009
  • Mathematica
    fStanley[n_Integer]:=Product[(1+q^(2i-1))/(1-q^(4i))/(1+q^(4i-2))^2, {i, n}]; Table[PartitionsP[n]/2+1/2*Coefficient[Series[fStanley[n], {q, 0, n+1}], q^n], {n, 64}] or Table[Count[Partitions[n], q_/;Mod[Count[q, w_/;OddQ[w]]- Count[TransposePartition[q], w_/;OddQ[w]], 4]===0], {n, 24}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^8] / (EllipticTheta[ 3, 0, x^2] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Jun 01 2014 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^16 + A)^5 / (eta(x + A) * eta(x^4 + A)^5 * eta(x^32 + A)^2), n))}; /* Michael Somos, May 04 2011 */

Formula

From Michael Somos, May 04 2011: (Start)
Expansion of q^(1/24) * eta(q^2)^2 * eta(q^16)^5 / (eta(q) * eta(q^4)^5 * eta(q^32)^2) in powers of q.
Expansion of phi(x^8) / (phi(x^2) * f(-x)) in powers of x where phi(), f() are Ramanujan theta functions.
Euler transform of period 32 sequence [ 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, -1, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 1, ...].
G.f.: theta_3(x^8) / (theta_3(x^2) * Product_{k>0} (1 - x^k)) = A000041(x) * A112128(x^2).
a(n) = (A000041(n) + A085261(n)) / 2.
(End)
Showing 1-2 of 2 results.

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