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A053691 Number of 11-core partitions of n.

%I A053691 #33 Aug 11 2025 08:43:57
%S A053691 1,1,2,3,5,7,11,15,22,30,42,45,66,79,102,121,154,176,220,248,297,330,
%T A053691 430,452,552,605,720,777,935,990,1182,1265,1485,1530,1838,1892,2214,
%U A053691 2310,2684,2750,3238,3289,3850,3960,4500,4599,5370,5404,6220,6325,7238
%N A053691 Number of 11-core partitions of n.
%C A053691 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A053691 Seiichi Manyama, <a href="/A053691/b053691.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A053691 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.
%H A053691 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A053691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A053691 Expansion of f(-x^11)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.
%F A053691 Expansion of q^-5 * etq(q^11)^11 / eta(q) in powers of q. - _Michael Somos_, Nov 06 2014
%F A053691 Euler transform of period 11 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, ...]. - _Michael Somos_, Nov 06 2014
%F A053691 G.f. Product_{k>0} (1 - x^(11*k))^11 / (1 - x^k).
%e A053691 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 + ...
%e A053691 G.f. = q^5 + q^6 + 2*q^7 + 3*q^8 + 5*q^9 + 7*q^10 + 11*q^11 + 15*q^12 + ...
%t A053691 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. *)
%t A053691 a[ n_] := SeriesCoefficient[ QPochhammer[ x^11]^11 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Nov 06 2014 *)
%o A053691 (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 */
%Y A053691 Column t=11 of A175595.
%K A053691 easy,nice,nonn
%O A053691 0,3
%A A053691 _James Sellers_, Feb 14 2000