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A101277
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Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.
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1
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1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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).
This is also A080054 times 1/product_{k>0}(1-x^(2k))
There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos Oct 27 2008
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REFERENCES
| Noureddine Chair, Partition Identities From Partial Supersymmetry, hep-th/0409011
Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
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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
| Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos Feb 10 2005
G.f.:=1/theta_4(0, x)product_{k>0}(1+x^(2k))= theta_4(0, x^2)/theta_4(0, x)product_{k>0}(1-x^(2k))= 1/product_{k>0}(1-x^(2k-1))^2(1-x^(4k)).
Expansion of 1 / (psi(-q) * chi(-q)) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos Oct 27 2008
Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos Oct 27 2008
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EXAMPLE
| E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.
1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + 38*q^83 + ...
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MAPLE
| series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))), k=1..100), x=0, 100);
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2/eta(x+A)^2/eta(x^4+A), n))} /* Michael Somos Feb 10 2005 */
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CROSSREFS
| Cf. A015128, A098151, A080054.
Sequence in context: A075623 A024801 A146163 * A201077 A023655 A023561
Adjacent sequences: A101274 A101275 A101276 * A101278 A101279 A101280
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KEYWORD
| nonn
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AUTHOR
| Noureddine Chair (n.chair(AT)rocketmail.com), Dec 20 2004; revised Jan 05 2005
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