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Search: id:A053692
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| A053692 |
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Number of self-conjugate 4-core partitions of n. |
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+0
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; listen)
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OFFSET
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0,11
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COMMENT
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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
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Garvan, F., Kim, D. and Stanton, D., Cranks and t-cores, Inventiones Math. 101 (1990) 1-17
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.
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FORMULA
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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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PROGRAM
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(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
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Cf. A053693.
Sequence in context: A050332 A029425 A025902 this_sequence A099494 A030341 A121444
Adjacent sequences: A053689 A053690 A053691 this_sequence A053693 A053694 A053695
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KEYWORD
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easy,nonn
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AUTHOR
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James A. Sellers (sellersj(AT)math.psu.edu), Feb 14 2000
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