A006353 - OEIS

%I M3825 #73 Aug 02 2024 22:38:58

%S 1,5,13,23,29,30,31,40,61,77,78,60,47,70,104,138,125,90,85,100,174,

%T 184,156,120,79,155,182,239,232,150,186,160,253,276,234,240,101,190,

%U 260,322,366,210,248,220,348,462,312,240,143,285,403,414,406,270

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

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C Expansion of a modular form related to Apery numbers A005259. - _Michael Somos_, Mar 25 1999

%C Number 11 and 33 of the 126 eta-quotients listed in Table 1 of Williams 2012. - _Michael Somos_, Nov 10 2018

%D 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.

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

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

%H Seiichi Manyama, <a href="/A006353/b006353.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H F. Beukers, <a href="http://dx.doi.org/10.1016/0022-314X(87)90025-4">Another congruence for the Apéry numbers</a>, J. Number Theory 25 (1987), no. 2, 201-210.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H K. S. Williams, <a href="http://dx.doi.org/10.1142/S1793042112500595">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

%F 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.

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

%F Euler transform of period 6 sequence [5, -2, -2, -2, 5, -4, ...]. - _Michael Somos_, Oct 11 2006

%F 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

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

%F 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 ]

%e 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 + ...

%t 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_ *)

%t 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 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3])^7 / (QPochhammer[ q] QPochhammer[ q^6])^5, {q, 0, n}]; (* _Michael Somos_, May 27 2014 *)

%o (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 */

%o (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 */

%o (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

%o (Sage) A = ModularForms( Gamma0(6), 2, prec=56).basis(); A[0] + 5*A[1] + 13*A[2]; # _Michael Somos_, Sep 04 2013

%o (Magma) A := Basis(ModularForms(Gamma0(6), 2)); PowerSeries( A[1] + 5*A[2] + 13*A[3], 56); /* _Michael Somos_, Sep 04 2013 */

%o (Ruby)

%o def A000203(n)

%o   s = 0

%o   (1..n).each{|i| s += i if n % i == 0}

%o   s

%o end

%o def A006353(n)

%o   a = [0] + (1..n).map{|i| A000203(i)}

%o   ary = [1]

%o   (1..n).each{|i|

%o     ary[i] = 5 * a[i]

%o     ary[i] -=  2 * a[i / 2] if i % 2 == 0

%o     ary[i] +=  3 * a[i / 3] if i % 3 == 0

%o     ary[i] -= 30 * a[i / 6] if i % 6 == 0

%o   }

%o   ary

%o end

%o p A006353(100) # _Seiichi Manyama_, Jun 09 2017

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

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Extended with PARI programs by _Michael Somos_