A053691 - OEIS

%I #29 Jun 16 2023 14:59:40

%S 1,1,2,3,5,7,11,15,22,30,42,45,66,79,102,121,154,176,220,248,297,330,

%T 430,452,552,605,720,777,935,990,1182,1265,1485,1530,1838,1892,2214,

%U 2310,2684,2750,3238,3289,3850,3960,4500,4599,5370,5404,6220,6325,7238

%N Number of 11-core partitions of n.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H 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 F. Garvan, D. Kim and D. Stanton, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00210752X">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of f(-x^11)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

%F Expansion of q^-5 * etq(q^11)^11 / eta(q) in powers of q. - _Michael Somos_, Nov 06 2014

%F Euler transform of period 11 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, ...]. - _Michael Somos_, Nov 06 2014

%F G.f. Product_{k>0} (1 - x^(11*k))^11 / (1 - x^k).

%e 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 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 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 a[ n_] := SeriesCoefficient[ QPochhammer[ x^11]^11 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Nov 06 2014 *)

%o (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 Column t=11 of A175595.

%K easy,nice,nonn

%O 0,3

%A _James A. Sellers_, Feb 14 2000