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

Author

Michael Somos, Mar 24 2003

Keywords

• nonn
• easy

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.

Examples

			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 + ...
		

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.

Crossrefs

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

Programs

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))}

Formula

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