|
%I #16 Oct 03 2017 15:38:16
%S 1,1,2,2,3,1,2,0,2,1,2,2,4,2,2,2,1,0,0,0,3,2,4,2,4,1,2,0,2,0,2,2,2,4,
%T 2,0,1,0,0,2,2,2,4,2,6,1,2,0,2,-1,2,2,4,2,0,2,0,0,0,0,4,2,4,2,3,2,4,0,
%U 2,2,2,2,2,2,2,2,0,0,0,0,1,1,4,2,6,0,2,0,4,0,0,2,4,4,2,0,2,0,0,2,3,2,4,2,4,2,2,0,0,0,4,2,0,2,0,2,0,0,0,-2,4,3,4,4,6,1,2
%N G.f.: A(x) = 1 + Sum_{n>=0} (-1)^n* Sum_{k=1..4} x^(5n+k)/(1-x^(5n+k)).
%C Records are A003586 (numbers of the form 2^i*3^j) = [1,2,3,4,6,8,9,12...], occurring at positions: [0,2,4,12,44,132,484,924,4092,...].
%H G. C. Greubel, <a href="/A119428/b119428.txt">Table of n, a(n) for n = 0..1000</a>
%H L.-C. Shen, <a href="https://doi.org/10.1090/S0002-9947-1994-1250827-3">On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5</a>, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 342, Eq. (3.40).
%F a(5n) = a(n); a(10n+2) = 2*a(5n+1).
%F Convolution identity: Sum_{k=0..n} a(5(n-k))*a(5k+3) = Sum_{k=0..n} a(5(n-k)+1)*a(5k+2).
%F Euler transform of period 10 sequence: [1,1,0,0,-2,0,0,1,1,-2]; also,
%F Moebius transform of period 10 sequence: [1,1,1,1,0,-1,-1,-1,-1,0].
%F Expansion of (f(-q^5) * f(-q^10))^2 / (f(-q, -q^9) * f(-q^2, -q^8)) in powers of q where f() is Ramanujan's two-variable theta function. - _Michael Somos_, Mar 08 2008
%F G.f.: Product_{k>0} (1 - x^(5*k))^2 / ( (1-x^(10*k-1)) * (1-x^(10*k-2)) * (1-x^(10*k-8)) * (1-x^(10*k-9)) ). - _Michael Somos_, Mar 08 2008
%F G.f.: 1 + Sum_{k>0} (x^k + x^(2*k) + x^(3*k) + x^(4*k)) / (1 + x^(5*k)). - _Michael Somos_, Mar 08 2008
%F If A = A0 + A1 + A2 + A3 + A4 is the 5-section, then A0 * A3 = A1 * A2, A0 * A2 + A3 * A4 = 2 * A1^2. - _Michael Somos_, Mar 08 2008
%e A(x) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + x^5 + 2*x^6 + 2*x^8 + x^9 +...
%e = 1 + x/(1-x) + x^2/(1-x^2) + x^3/(1-x^3) + x^4/(1-x^4) +
%e - x^6/(1-x^6) - x^7/(1-x^7) - x^8/(1-x^8) - x^9/(1-x^9) +
%e + x^11/(1-x^11) + x^12/(1-x^12) + x^13/(1-x^13) + x^14/(1-x^14) +...
%t A119428[n_] := SeriesCoefficient[(QPochhammer[q^5,q^5])^2/(QPochhammer[q, q^10]*QPochhammer[q^2, q^10]*QPochhammer[q^8, q^10]*QPochhammer[q^9, q^10]), {q, 0, n}]; Table[A119428[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 03 2017 *)
%o (PARI) {a(n)=polcoeff(1+sum(k=0,n\5+1,(-1)^k*sum(j=1,4,x^(5*k+j)/(1-x^(5*k+j)+x*O(x^n))) ),n)}
%o (PARI) {a(n) = local(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k+ x*O(x^n))^[2,-1,-1,0,0,2,0,0,-1,-1][k%10 + 1]), n))} /* _Michael Somos_, Mar 08 2008 */
%Y Cf. A113973.
%K sign
%O 0,3
%A _Paul D. Hanna_, May 19 2006
|
