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Search: id:A131945
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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; listen)
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OFFSET
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0,3
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COMMENT
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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.yu), Aug 01 2007
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LINKS
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Brian Drake, Table of n, a(n) for n = 0..100
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FORMULA
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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
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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
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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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PROGRAM
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(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
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Cf. A006950, A131942.
Sequence in context: A069906 A053097 A035946 this_sequence A035951 A035957 A035964
Adjacent sequences: A131942 A131943 A131944 this_sequence A131946 A131947 A131948
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KEYWORD
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easy,nonn
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AUTHOR
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Brian Drake (bdrake(AT)brandeis.edu), Jul 30 2007
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