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

0,3

Comments

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

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

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).

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Column t=11 of A175595.

Sequence in context: A261776 A027344 A184645 * A242696 A218510 A026816

Adjacent sequences: A053688 A053689 A053690 * A053692 A053693 A053694

Keyword

easy,nice,nonn

Author

James A. Sellers, Feb 14 2000

Status

approved