A045831 Number of 4-core partitions of n.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Conjecturally Sum_n a(n)q^(8n+5) equals theta series of sodalite. - Fred Lunnon, Mar 05 2015

Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - Michael Somos, Feb 10 2020

Examples

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

References

• 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.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• M. Hirschhorn, and J. Sellers, Some amazing facts about 4-cores, J. Num. Thy. 60 (1996), 51-69.
• K. Ono, and L. Sze, 4-core partitions and class numbers, Acta. Arith. 80 (1997), 249-272.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

A004024/4, column t=4 of A175595.

Cf. A286953.

Cf. A008443, A045818, A213624.

Programs

Mathematica

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 *)

PARI

{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 */

Formula

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.

From Michael Somos, Mar 24 2003: (Start)

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

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

(End)

Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006

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

Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2008

Extensions

More terms from James Sellers, Feb 11 2000