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A108961
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Number of partitions that are "2-close" to being self-conjugate.
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1
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1, 1, 2, 3, 3, 5, 7, 9, 12, 16, 20, 26, 33, 41, 51, 64, 79, 97, 119, 144, 175, 212, 254, 305, 365, 434, 516, 612, 722, 851, 1002, 1174, 1375, 1607, 1872, 2179, 2531, 2933, 3395, 3923, 4524, 5211, 5994, 6881, 7891, 9038, 10334, 11804, 13467, 15341, 17460, 19849
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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).
Let (a1,a2,a3,...ad:b1,b2,b3,...bd) be the Frobenius symbol for a partition pi of n. Then pi is m-close to being self-conjugate if for all k, |ak-bk| <= m.
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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
| Define the Dedekind eta function = q^1/24. Product(1-q^k), k >=1. Then the number of m-close partitions is q^(1/24).(m+2)^2/(1.(2m+4)) (where m denotes eta(q^m)).
Expansion of q^(1/24)*eta(q^4)^2/(eta(q)*eta(q^8)) in powers of q. - Michael Somos Oct 17 2006
Expansion of chi(q^2)*chi(-q) in powers of q where chi() is a Ramanujan theta function. - Michael Somos Oct 17 2006
Euler transform of period 8 sequence [ 1, 1, 1, -1, 1, 1, 1, 0, ...]. - Michael Somos Oct 17 2006
G.f.: Product_{k>0} (1+x^k)*(1+x^(2k))/(1+x^(4k)). - Michael Somos Oct 17 2006
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^2/ (eta(x+A)*eta(x^8+A)), n))} /* Michael Somos Oct 17 2006 */
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CROSSREFS
| Cf. A000700 for m=0 (self-conjugate), A070047 for m=1, A108962 for m=3.
Sequence in context: A111865 A042955 A035553 * A017984 A035066 A035068
Adjacent sequences: A108958 A108959 A108960 * A108962 A108963 A108964
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KEYWORD
| nonn
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AUTHOR
| John McKay (mckay(AT)cs.concordia.ca), Jul 22 2005
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