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A056970
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Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.
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0
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1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 13, 13, 15, 16, 17, 20, 21, 23, 25, 27, 30, 33, 36, 38, 42, 45, 49, 54, 57, 62, 67, 72, 79, 85, 92, 98, 106, 114, 123, 133, 141, 152, 163, 175, 189, 202, 216, 231, 248, 265, 284, 304, 323
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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COMMENTS
| Also number of partitions of n into parts equal to 2,5, or 11 mod 12 (Gollnitz's theorem). Example: a(18)=4 because we have [14,2,2],[11,5,2],[5,5,2,2,2,2] and [2,2,2,2,2,2,2,2,2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
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REFERENCES
| G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 101.
H. Goellnitz, Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. Vol. 225 (1967), 154-190.
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LINKS
| K. Alladi, Going beyond the partition theorem of Goellnitz
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| G.f.=product((1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), j=0..infinity). G.f.=1/product((1-x^(2+12j))(1-x^(5+12j))(1-x^(11+12j)),j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
Euler transform of period 12 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos Jul 24 2007
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EXAMPLE
| a(18)=4 because we have [16,2],[14,4],[11,5,2] and [10,8].
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MAPLE
| g:=product((1+x^(2+6*j))*(1+x^(4+6*j))*(1+x^(5+6*j)), j=0..30): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..67); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
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MATHEMATICA
| max = 70; g[x_] := Product[(1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), {j, 0, Floor[max/6]}]; CoefficientList[ Series[g[x], {x, 0, max}], x](* From Jean-François Alcover, Nov 16 2011, after Emeric Deutsch *)
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PROG
| (PARI) {a(n)= if(n<0, 0, polcoeff( 1/prod(k=1, n, 1-(k%3==2)*(k%12!=8)*x^k, 1+x*O(x^n)), n))} /* Michael Somos Jul 24 2007 */
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CROSSREFS
| Sequence in context: A194328 A194304 A090701 * A008668 A116563 A076695
Adjacent sequences: A056967 A056968 A056969 * A056971 A056972 A056973
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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