A104501 - OEIS

%I #18 Apr 09 2018 02:52:59

%S 1,1,2,3,5,7,11,15,22,30,42,56,76,100,133,172,225,288,371,470,598,751,

%T 945,1177,1468,1815,2245,2757,3386,4133,5043,6121,7425,8966,10818,

%U 13001,15610,18677,22324,26600,31662,37582,44560,52701,62261,73387,86406

%N Coefficients of the A-Dyson Mod 27 identity.

%H G. C. Greubel, <a href="/A104501/b104501.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^12,-q^15)/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= A1^2*A4^2 +2*A2^2*A3^2 -A1*A3^3 -A4*A2^3 -A1*A2*A3*A4.

%F G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-12))(1-x^(27k-15))/(1-x^k).

%F G.f.: 1+ Sum_{k>0} x^k^2 ( Product_{j=1..k-1} 1-x^(3j) )/( (Product_{j=1..2k-1} (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 +11*q^6 +15*q^7 +22*q^8 +30*q^9 +...

%t QP = QPochhammer; (QP[1/x^15, x^27]*QP[1/x^12, x^27]*QP[x^27])/((1-1/x^15)* (1-1/x^12)*QP[x]) + O[x]^50 // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 18 2017 *)

%o (PARI) {a(n)=local(m); if(n<0, 0, m=sqrtint(24*n+1); polcoeff( sum(k= -((m-1)\18), (m+1)\18, (-1)^k*x^((9*k^2-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<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2* prod(j=1, k-1, (1-x^(3*j))/(1-x^(j+1))/(1-x^(2*j))/(1-x^(2*j+1)), 1+O(x^(n-k^2+1)))/(1-x)^2, 1), n))} /* _Michael Somos_, Mar 15 2006 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = eta(x + x*O(x^n)) ; polcoeff( sum(k=0, n, (k%3==0) * polcoeff(A, k) * x^k) / A, n))} /* _Michael Somos_, Sep 29 2007 */

%Y Cf. A104502, A104503, A104504.

%K nonn

%O 0,3

%A _Eric W. Weisstein_, Mar 11 2005

%E Edited by _Michael Somos_, Mar 15 2006