login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A053693
Number of self-conjugate 8-core partitions of n.
1
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
0,9
LINKS
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
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)
EXAMPLE
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 + ...
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 *)
PROG
(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 */
CROSSREFS
Cf. A053692.
Sequence in context: A363822 A054893 A090617 * A330324 A341132 A291309
KEYWORD
easy,nonn
AUTHOR
James A. Sellers, Feb 14 2000
STATUS
approved