login
A104504
Coefficients of the D-Dyson mod 27 identity.
4
1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 782, 955, 1179, 1432, 1755, 2122, 2583, 3109, 3762, 4510, 5427, 6480, 7760, 9231, 11004, 13043, 15485, 18293, 21634, 25475, 30021, 35245, 41396, 48459, 56740
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities
FORMULA
Expansion of f(-q^3,-q^24)/f(-q,-q^2) in powers of q where f() is Ramanujan's theta function.
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.
G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-3))(1-x^(27k-24))/(1-x^k).
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)) ).
A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos, Sep 29 2007
EXAMPLE
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 + ...
MATHEMATICA
QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y];
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 *)
PROG
(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 */
(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 */
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Mar 11 2005
EXTENSIONS
Edited by Michael Somos, Mar 15 2006
STATUS
approved