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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) (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 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 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 EXAMPLE 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 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 *) PROG {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 */ CROSSREFS A004024/4, column t=4 of A175595. Cf. A286953. Cf. A008443, A045818, A213624. Sequence in context: A239214 A200181 A121062 * A046821 A239691 A265496 Adjacent sequences:  A045828 A045829 A045830 * A045832 A045833 A045834 KEYWORD nonn AUTHOR EXTENSIONS More terms from James A. Sellers, Feb 11 2000 STATUS approved

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Last modified March 31 22:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)