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A045831 Number of 4-core partitions of n. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

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

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

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

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.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

EXAMPLE

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]

MATHEMATICA

QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Jul 26 2011, updated Nov 29 2015 *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^4/eta(x+A), n))}

CROSSREFS

A004024/4, column t=4 of A175595.

Cf. A286953.

Sequence in context: A239214 A200181 A121062 * A046821 A239691 A265496

Adjacent sequences:  A045828 A045829 A045830 * A045832 A045833 A045834

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms and additional comments from James A. Sellers, Feb 11 2000

STATUS

approved

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Last modified November 15 08:37 EST 2019. Contains 329144 sequences. (Running on oeis4.)