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A104504 Coefficients of the D-Dyson mod 27 identity.

%I A104504 #14 Feb 16 2025 08:32:56
%S A104504 1,1,2,2,4,5,8,10,15,19,27,34,47,59,79,99,130,162,209,259,330,407,512,
%T A104504 628,782,955,1179,1432,1755,2122,2583,3109,3762,4510,5427,6480,7760,
%U A104504 9231,11004,13043,15485,18293,21634,25475,30021,35245,41396,48459,56740
%N A104504 Coefficients of the D-Dyson mod 27 identity.
%H A104504 G. C. Greubel, <a href="/A104504/b104504.txt">Table of n, a(n) for n = 0..1000</a>
%H A104504 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DysonMod27Identities.html">Dyson Mod 27 Identities</a>
%F A104504 Expansion of f(-q^3,-q^24)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
%F A104504 Given A=A0+A1+A2+A3+A4 is the 5-section, then 0= 2*A0^2*A1^2 +A2^2*A4^2 -A2*A0^3 -A4*A1^3 -A0*A1*A2*A4.
%F A104504 G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-3))(1-x^(27k-24))/(1-x^k).
%F A104504 G.f.: Sum_{k>0} x^(k^2+3k) ( Product_{j=1..k} 1-x^(3j) )/ ( (Product_{j=1..2k+2} (1-x^j)) (Product_{j=1..k}(1-x^j)) ).
%F A104504 A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - _Michael Somos_, Sep 29 2007
%e A104504 1 + q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 8*q^6 + 10*q^7 + 15*q^8 + 19*q^9 + ...
%t A104504 QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y];
%t A104504 a[n_] := SeriesCoefficient[f[-q^3, -q^24]/f[-q, -q^2], {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 08 2018 *)
%o A104504 (PARI) {a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+49); polcoeff( sum(k= -((m-7)\18), (m+7)\18, (-1)^k*x^((9*k^2-7*k)*3/2),x*O(x^n))/ eta(x+x*O(x^n)), n))} /* _Michael Somos_, Mar 15 2006 */
%o A104504 (PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=0, sqrtint(n+1)-1, x^(k^2+3*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 A104504 (PARI) {a(n) = local(A); if( n<0, 0, n+=2; A = eta(x + x*O(x^n)) ; polcoeff( - sum(k=0, n, (k%3==2) * polcoeff(A, k) * x^k) / A, n))} /* _Michael Somos_, Sep 29 2007 */
%Y A104504 Cf. A104501, A104502, A104503.
%K A104504 nonn
%O A104504 0,3
%A A104504 _Eric W. Weisstein_, Mar 11 2005
%E A104504 Edited by _Michael Somos_, Mar 15 2006