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A104503 Coefficients of the C-Dyson Mod 27 identity. 4
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 189, 241, 308, 389, 492, 615, 770, 956, 1187, 1463, 1802, 2207, 2701, 3288, 3999, 4842, 5857, 7056, 8491, 10183, 12197, 14564, 17369, 20658, 24539, 29075, 34408, 40627, 47912, 56385, 66277 (list; graph; refs; listen; history; text; internal format)
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^6,-q^21)/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= A0^2*A3^2 +2*A1^2*A2^2 -A0*A2^3 -A3*A1^3 -A0*A1*A2*A3.

G.f.: Product_{k>0} (1-x^(27k))(1-x^(27k-6))(1-x^(27k-21))/(1-x^k).

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)) ).

A104501(n) = A104503(n-1) + A104504(n-2) unless n=0. - Michael Somos, Sep 29 2007

EXAMPLE

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 + ...

MATHEMATICA

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 *)

PROG

(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 */

(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 */

(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 */

CROSSREFS

Cf. A104501, A104502, A104504.

Sequence in context: A035984 A035994 A036005 * A027340 A000701 A123975

Adjacent sequences:  A104500 A104501 A104502 * A104504 A104505 A104506

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Mar 11 2005

STATUS

approved

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Last modified July 10 15:45 EDT 2020. Contains 335577 sequences. (Running on oeis4.)