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A053692
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Number of self-conjugate 4-core partitions of n.
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6
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1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 3, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 3, 1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 1, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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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) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Euler transform of period 8 sequence [1,-1,1,0,1,-1,1,-2,...].
Expansion of q^(-5/8)eta(q^2)^2eta(q^8)^2/(eta(q)eta(q^4)) in powers of q.
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.
Garvan, F., Kim, D. and Stanton, D., Cranks and t-cores, Inventiones Math. 101 (1990) 1-17
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: product((1-q^(8*i))^2*(1-q^(4*i-2))/(1-q^(2*i-1)), i=1..inifinity)
G.f.: psi(x)psi(x^4) where psi(x) is a Ramanujan theta function. - Michael Somos Nov 03 2005
G.f.: Sum_{k} x^k/(1-x^(8k+5)). - Michael Somos Nov 03 2005
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=sum(k=0, ceil(sqrtint(8*n+1)/2), x^((k^2+k)/2), x*O(x^n)); polcoeff( A*subst(A+x*O(x^(n\4)), x, x^4), n))} /* Michael Somos Nov 03 2005 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^8+A)^2/ eta(x+A)/eta(x^4+A), n))}
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CROSSREFS
| Cf. A053693.
Sequence in context: A050332 A029425 A025902 * A099494 A030341 A121444
Adjacent sequences: A053689 A053690 A053691 * A053693 A053694 A053695
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KEYWORD
| easy,nonn
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AUTHOR
| James A. Sellers (sellersj(AT)math.psu.edu), Feb 14 2000
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