|
%I #17 Apr 09 2018 02:52:36
%S 1,1,2,3,5,7,10,14,20,27,37,49,66,86,113,146,189,241,308,389,492,615,
%T 770,956,1187,1463,1802,2207,2701,3288,3999,4842,5857,7056,8491,10183,
%U 12197,14564,17369,20658,24539,29075,34408,40627,47912,56385,66277
%N Coefficients of the C-Dyson Mod 27 identity.
%H G. C. Greubel, <a href="/A104503/b104503.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DysonMod27Identities.html">Dyson Mod 27 Identities</a>
%F Expansion of f(-q^6,-q^21)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
%F Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= A0^2*A3^2 +2*A1^2*A2^2 -A0*A2^3 -A3*A1^3 -A0*A1*A2*A3.
%F G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-6))(1-x^(27k-21))/(1-x^k).
%F G.f.: Sum_{k>0} x^(k^2+2k) ( Product_{j=1..k} 1-x^(3j) )/ ( (Product_{j=1..2k+2} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
%F A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - _Michael Somos_, Sep 29 2007
%e 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 10*q^6 + 14*q^7 + 20*q^8 + 27*q^9 + ...
%t QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; a[n_]:= SeriesCoefficient[f[-q^12, -q^15]/f[-q, -q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Apr 08 2018 *)
%o (PARI) {a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+25); polcoeff( sum(k= -((m-5)\18), (m+5)\18, (-1)^k*x^((9*k^2-5*k)*3/2),x*O(x^n))/ eta(x+x*O(x^n)), n))} /* _Michael Somos_, Mar 15 2006 */
%o (PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=0, sqrtint(n+1)-1, x^(k^2+2*k)* prod(j=1, k, (1-x^(3*j))/(1-x^j)/(1-x^(2*j+1))/(1-x^(2*j+2)), 1+O(x^(n-k^2-2*k+1)))/(1-x)/(1-x^2) ), n))} /* _Michael Somos_, Mar 15 2006 */
%o (PARI) {a(n) = local(A); if( n<0, 0, n++; A = eta(x + x*O(x^n)) ; polcoeff( - sum(k=0, n, (k%3==1) * polcoeff(A, k) * x^k) / A, n))} /* _Michael Somos_, Sep 29 2007 */
%Y Cf. A104501, A104502, A104504.
%K nonn
%O 0,3
%A _Eric W. Weisstein_, Mar 11 2005
|
