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A104501 Coefficients of the A-Dyson Mod 27 identity. 4
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 172, 225, 288, 371, 470, 598, 751, 945, 1177, 1468, 1815, 2245, 2757, 3386, 4133, 5043, 6121, 7425, 8966, 10818, 13001, 15610, 18677, 22324, 26600, 31662, 37582, 44560, 52701, 62261, 73387, 86406 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..46.

Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

FORMULA

Expansion of f(-q^12,-q^15)/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= A1^2*A4^2 +2*A2^2*A3^2 -A1*A3^3 -A4*A2^3 -A1*A2*A3*A4.

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

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)) ).

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 +11*q^6 +15*q^7 +22*q^8 +30*q^9 +...

PROG

(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 */

(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 */

(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 */

CROSSREFS

Cf. A104502, A104503, A104504.

Sequence in context: A008634 A238869 A036011 * A242697 A218512 A008635

Adjacent sequences:  A104498 A104499 A104500 * A104502 A104503 A104504

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Mar 11 2005

EXTENSIONS

Edited by Michael Somos, Mar 15 2006

STATUS

approved

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Last modified November 24 04:08 EST 2014. Contains 249867 sequences.