A045831 - OEIS

%I #59 Nov 23 2021 05:48:50

%S 1,1,2,3,1,3,3,3,4,4,2,2,7,3,5,6,2,4,7,3,4,7,5,8,5,4,4,8,5,6,7,2,9,11,

%T 3,8,9,4,6,5,7,5,14,7,4,10,5,10,11,3,9,10,5,8,10,4,6,15,8,9,10,6,8,15,

%U 6,10,6,5,15,9,6,8,14,8,6,13,5,16,18,7,8,7,9,6,15,6,12,17,5,8,15,7,12

%N Number of 4-core partitions of n.

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

%C Conjecturally  Sum_n a(n)q^(8n+5) equals theta series of sodalite. - _Fred Lunnon_, Mar 05 2015

%C Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - _Michael Somos_, Feb 10 2020

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. II, p. 23.

%H Seiichi Manyama, <a href="/A045831/b045831.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H M. Hirschhorn, and J. Sellers, <a href="http://dx.doi.org/10.1006/jnth.1996.0112">Some amazing facts about 4-cores</a>, J. Num. Thy. 60 (1996), 51-69.

%H K. Ono, and L. Sze, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa80/aa8035.pdf">4-core partitions and class numbers</a>, Acta. Arith. 80 (1997), 249-272.

%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 eta(32*z)^4/eta(8*z) = Sum_{x, y, z} q^(x^2+2*y^2+2*z^2), x, y, z >= 1 and odd.

%F From _Michael Somos_, Mar 24 2003: (Start)

%F Euler transform of period 4 sequence [1, 1, 1, -3, ...].

%F Expansion of q^(-5/8) * eta(q^4)^4/eta(q) in powers of q.

%F (End)

%F Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - _Michael Somos_, Jan 02 2006

%F G.f.: Product_{k>0} (1-q^(4*k))^4/(1-q^k).

%F Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Sep 02 2008

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...

%e G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ... ,

%e apparently the theta series of the sodalite net, aka edge-skeleton of space honeycomb by truncated octahedra. - _Fred Lunnon_, Mar 05 2015

%t QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 26 2011, updated Nov 29 2015 *)

%o (PARI)

%o {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 / eta(x + A), n))}; /* _Michael Somos_, Mar 24 2003 */

%Y A004024/4, column t=4 of A175595.

%Y Cf. A286953.

%Y Cf. A008443, A045818, A213624.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Feb 11 2000