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

%I A053693 #16 Aug 11 2025 08:42:50
%S A053693 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,
%T A053693 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,
%U A053693 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
%N A053693 Number of self-conjugate 8-core partitions of n.
%H A053693 F. Garvan, D. Kim and D. Stanton, <a href="https://gdz.sub.uni-goettingen.de/id/PPN356556735_0101">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.
%F A053693 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
%F A053693 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
%F A053693 G.f.: product((1-q^(16*i))^4*(1-q^(4*i-2))/(1-q^(2*i-1)), i=1..infinity)
%e A053693 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 + ...
%e A053693 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 + ...
%t A053693 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ x^16]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%o A053693 (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 */
%Y A053693 Cf. A053692.
%K A053693 easy,nonn
%O A053693 0,9
%A A053693 _James Sellers_, Feb 14 2000