A104504 Coefficients of the D-Dyson mod 27 identity.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Cf. A104501, A104502, A104503.

Sequence in context: A035981 A035991 A036002 * A027337 A364892 A325434

Adjacent sequences: A104501 A104502 A104503 * A104505 A104506 A104507

Keyword

nonn

Author

Eric W. Weisstein, Mar 11 2005

Extensions

Edited by Michael Somos, Mar 15 2006

Status

approved