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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)
6
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 (list; graph; refs; listen; history; text; internal format)
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.
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
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Extended with PARI programs by Michael Somos
STATUS
approved

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Last modified February 27 12:16 EST 2024. Contains 370375 sequences. (Running on oeis4.)