cp's OEIS Frontend

A081622 Number of 6-core partitions of n.

%I A081622 #20 May 19 2017 07:41:46
%S A081622 1,1,2,3,5,7,5,9,10,12,12,14,20,20,21,23,24,24,32,29,35,36,44,47,38,
%T A081622 47,49,52,55,58,59,64,66,71,70,78,79,88,87,90,85,87,111,104,102,107,
%U A081622 112,113,121,113,130,130,148,153,132,147,149,156,162,149,167,160,178,180
%N A081622 Number of 6-core partitions of n.
%C A081622 Euler transform of period 6 sequence [ 1, 1, 1, 1, 1, -5, ...].
%C A081622 Expansion of q^(-35/24) * eta(q^6)^6 / eta(q) in powers of q.
%H A081622 Seiichi Manyama, <a href="/A081622/b081622.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A081622 F. Garvan, D. Kim and D. Stanton, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN00210752X&amp;IDDOC=178508">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.
%F A081622 G.f.: Product_{k>0} (1 - x^(6*k))^6 / (1 - x^k).
%e A081622 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 5*x^6 + 9*x^7 + 10*x^8 + 12*x^9 + ...
%e A081622 q^35 + q^59 + 2*q^83 + 3*q^107 + 5*q^131 + 7*q^155 + 5*q^179 + 9*q^203 + ...
%o A081622 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^(6*k) + x * O(x^n))^6 / (1 - x^k)), n))}
%o A081622 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / eta(x + A), n))}
%Y A081622 Cf. A010054, A033687, A045831, A053723, A053724.
%K A081622 nonn,easy
%O A081622 0,3
%A A081622 _Michael Somos_, Mar 24 2003