A006353 Expansion of (phi(-q^3) * psi(q))^3 / (phi(-q) * psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions. (Formerly M3825)

1, 5, 13, 23, 29, 30, 31, 40, 61, 77, 78, 60, 47, 70, 104, 138, 125, 90, 85, 100, 174, 184, 156, 120, 79, 155, 182, 239, 232, 150, 186, 160, 253, 276, 234, 240, 101, 190, 260, 322, 366, 210, 248, 220, 348, 462, 312, 240, 143, 285, 403, 414, 406, 270

Offset

0,2

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).

Expansion of a modular form related to Apery numbers A005259. - Michael Somos, Mar 25 1999

Number 11 and 33 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

References

M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Zagier, "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Formula

Expansion of (b(q^2)^2 / b(q)) * (c(q)^2 / c(q^2)) / 3 in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 in powers of q.

Euler transform of period 6 sequence [5, -2, -2, -2, 5, -4, ...]. - Michael Somos, Oct 11 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 6 (t / i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 04 2013

G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 + x^k)^7 / (1 + x^(3*k))^5.

G.f.: Sum_{n>=0} A005259(n)*t(q)^n where t(q) = (eta(q)*eta(q^6)/(eta(q^2)*eta(q^3))^12. - Seiichi Manyama, Jun 10 2017 [See the Kontsevich-Zagier paper, section 2.4., and t is given in A226235.] - Wolfdieter Lang, May 16 2018 ]

Example

G.f. = 1 + 5*q + 13*q^2 + 23*q^3 + 29*q^4 + 30*q^5 + 31*q^6 + 40*q^7 + 61*q^8 + ...

Mathematica

EulerTransform[ seq_List ] := With[ {m = Length[seq]}, CoefficientList[ Series[ Times @@ (1/(1 - x^Range[m])^seq), {x, 0, m}], x]]; s6 = Table[ {5, -2, -2, -2, 5, -4}, {10}] // Flatten; EulerTransform[ s6 ] (* Jean-François Alcover, Mar 15 2012, after Michael Somos *)
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ d {0, 5, 4, 6, 4, 5}[[ Mod[d, 6] + 1]], {d, Divisors@n}]]; (* Michael Somos, May 27 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3])^7 / (QPochhammer[ q] QPochhammer[ q^6])^5, {q, 0, n}]; (* Michael Somos, May 27 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, d*[0, 5, 4, 6, 4, 5][ d%6 + 1]))}; /* Michael Somos, Oct 11 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A))^7 / (eta(x + A) * eta(x^6 + A))^5, n))}; /* Michael Somos, Oct 11 2006 */
(PARI) q='q+O('q^99); Vec((eta(q^2)*eta(q^3))^7/(eta(q)*eta(q^6))^5) \\ Altug Alkan, May 16 2018
(Sage) A = ModularForms( Gamma0(6), 2, prec=56).basis(); A[0] + 5*A[1] + 13*A[2]; # Michael Somos, Sep 04 2013
(Magma) A := Basis(ModularForms(Gamma0(6), 2)); PowerSeries( A[1] + 5*A[2] + 13*A[3], 56); /* Michael Somos, Sep 04 2013 */
(Ruby)
def A000203(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A006353(n)
  a = [0] + (1..n).map{|i| A000203(i)}
  ary = [1]
  (1..n).each{|i|
    ary[i] = 5 * a[i]
    ary[i] -=  2 * a[i / 2] if i % 2 == 0
    ary[i] +=  3 * a[i / 3] if i % 3 == 0
    ary[i] -= 30 * a[i / 6] if i % 6 == 0
  }
  ary
end
p A006353(100) # Seiichi Manyama, Jun 09 2017

Crossrefs

Cf. A000203, A005259, A006352 (E_2), A226235 (t(q)).

Sequence in context: A060004 A174172 A076408 * A155142 A155552 A219546

Adjacent sequences: A006350 A006351 A006352 * A006354 A006355 A006356

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Extended with PARI programs by Michael Somos

Status

approved