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.
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
Examples
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 + ...
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.
Programs
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Magma
A := Basis(ModularForms(Gamma0(6), 2)); PowerSeries( A[1] + 5*A[2] + 13*A[3], 56); /* Michael Somos, Sep 04 2013 */
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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 *) -
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 -
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
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Sage
A = ModularForms( Gamma0(6), 2, prec=56).basis(); A[0] + 5*A[1] + 13*A[2]; # Michael Somos, Sep 04 2013
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 ]
Extensions
Extended with PARI programs by Michael Somos
Comments