A053691 Number of 11-core partitions of n.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 45, 66, 79, 102, 121, 154, 176, 220, 248, 297, 330, 430, 452, 552, 605, 720, 777, 935, 990, 1182, 1265, 1485, 1530, 1838, 1892, 2214, 2310, 2684, 2750, 3238, 3289, 3850, 3960, 4500, 4599, 5370, 5404, 6220, 6325, 7238

Offset: 0

Author

James Sellers, Feb 14 2000

Keywords

• easy
• nice
• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^5 + q^6 + 2*q^7 + 3*q^8 + 5*q^9 + 7*q^10 + 11*q^11 + 15*q^12 + ...
		

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.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Column t=11 of A175595.

Programs

Mathematica

m = 50; CoefficientList[ Series[ Product[(1-q^(11*k))^11/(1-q^k), {k, 1, m}], {q, 0, m}], q] (* Jean-François Alcover, Jul 26 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^11]^11 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 06 2014 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^11 + A)^11 / eta(x + A), n))}; /* Michael Somos, Nov 06 2014 */

Formula

Expansion of f(-x^11)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

Expansion of q^-5 * etq(q^11)^11 / eta(q) in powers of q. - Michael Somos, Nov 06 2014

Euler transform of period 11 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, ...]. - Michael Somos, Nov 06 2014

G.f. Product_{k>0} (1 - x^(11*k))^11 / (1 - x^k).