A053693 Number of self-conjugate 8-core partitions of n.

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 1, 1, 5, 2, 3, 4, 4, 5, 3, 4, 4, 6, 4, 5, 6, 4, 5, 7, 6, 7, 7, 5, 7, 7, 6, 5, 8, 5, 5, 6, 6, 6, 13, 11, 4, 11, 7, 9, 9, 6, 11, 12, 10, 8, 13, 9, 8, 15, 9, 7, 12, 8, 10, 14, 9, 10, 13, 13, 8, 16, 12, 12, 15, 8, 9, 14, 12, 11, 19, 11, 12, 18, 14, 11, 17

Offset: 0

Author

James Sellers, Feb 14 2000

Keywords

• easy
• nonn

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2^x*10 + 2*x^11 + ...
G.f. = q^21 + q^29 + q^45 + q^53 + q^61 + q^69 + q^77 + 2*q^85 + 2*q^93 + 2*q^101 + ...
		

Links

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

Crossrefs

Cf. A053692.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ x^16]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^16 + A)^4 / (eta(x + A) * eta(x^4 + A)), n))}; /* Michael Somos, Apr 28 2003 */

Formula

Euler transform of period 16 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, -4, ...]. - Michael Somos, Apr 28 2003

Expansion of q^(-21/8) * eta(q^2)^2 * eta(q^16)^4 / (eta(q) * eta(q^4)) in powers of q. - Michael Somos, Apr 28 2003

G.f.: product((1-q^(16*i))^4*(1-q^(4*i-2))/(1-q^(2*i-1)), i=1..infinity)