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A131945
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Number of partitions of n where odd parts are distinct or repeated once.
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1
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1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 45, 55, 74, 90, 119, 145, 188, 228, 291, 351, 442, 532, 664, 796, 982, 1172, 1435, 1708, 2076, 2462, 2972, 3512, 4214, 4966, 5929, 6965, 8272, 9688, 11457, 13383, 15762, 18362, 21543, 25031, 29264, 33922, 39533, 45717
(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).
Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is product_{n=1..inf} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is product_{n=1..inf} (1-q^((2*m+2)*n))/(1-q^n). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 01 2007
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REFERENCES
| Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
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LINKS
| Brian Drake, Table of n, a(n) for n = 0..100
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: product_{n=1..inf} (1-q^(6n-3))/(1-q^n)
Expansion of chi(-q^3)/ f(-q) in powers of q where f(), chi() are Ramanujan theta functions. - Michael Somos Aug 05 2007
Expansion of q^(1/6)* eta(q^3)/ (eta(q)* eta(q^6)) in powers of q. - Michael Somos Aug 05 2007
Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos Aug 05 2007
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EXAMPLE
| a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
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MAPLE
| A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20), q, 21): seq(coeff(A, q, i), i=0..20);
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PROG
| (PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x^3+A)/ eta(x+A)/ eta(x^6+A), n))} /* Michael Somos Aug 05 2007 */
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CROSSREFS
| Cf. A006950, A131942.
Sequence in context: A183564 A053097 A035946 * A035951 A035957 A035964
Adjacent sequences: A131942 A131943 A131944 * A131946 A131947 A131948
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KEYWORD
| easy,nonn
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AUTHOR
| Brian Drake (bdrake(AT)brandeis.edu), Jul 30 2007
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