A081622 Number of 6-core partitions of n.

1, 1, 2, 3, 5, 7, 5, 9, 10, 12, 12, 14, 20, 20, 21, 23, 24, 24, 32, 29, 35, 36, 44, 47, 38, 47, 49, 52, 55, 58, 59, 64, 66, 71, 70, 78, 79, 88, 87, 90, 85, 87, 111, 104, 102, 107, 112, 113, 121, 113, 130, 130, 148, 153, 132, 147, 149, 156, 162, 149, 167, 160, 178, 180

Offset

0,3

Comments

Euler transform of period 6 sequence [ 1, 1, 1, 1, 1, -5, ...].

Expansion of q^(-35/24) * eta(q^6)^6 / eta(q) in powers of q.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Formula

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

Example

1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 5*x^6 + 9*x^7 + 10*x^8 + 12*x^9 + ...
q^35 + q^59 + 2*q^83 + 3*q^107 + 5*q^131 + 7*q^155 + 5*q^179 + 9*q^203 + ...

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^(6*k) + x * O(x^n))^6 / (1 - x^k)), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / eta(x + A), n))}

Crossrefs

Cf. A010054, A033687, A045831, A053723, A053724.

Sequence in context: A234316 A284630 A345872 * A064143 A283593 A115274

Adjacent sequences: A081619 A081620 A081621 * A081623 A081624 A081625

Keyword

nonn,easy

Author

Michael Somos, Mar 24 2003

Status

approved