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A001977
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Number of partitions of 3n into n parts from the set {0, 1, ..., 6} (repetitions admissible).
(Formerly M3335 N1342)
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6
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1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137, 2739, 3486, 4370, 5444, 6698, 8196, 9926, 11963, 14293, 17002, 20076, 23612, 27594, 32134, 37212, 42955, 49341, 56512, 64444, 73294, 83036, 93844, 105690, 118765, 133037
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OFFSET
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0,3
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COMMENTS
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In Cayley's terminology, this is the number of literal terms of degree n and weight 3*n involving the letters a, b, c, d, e, f, g, having weights 0, 1, 2, 3, 4, 5, 6 respectively, a number which is also equal to the coefficient of x^(3n)z^n in the development of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
a(0..5)=0; a(n) is the number of partitions of 3*(n+1) with 6 different numbers from the set {1,...,n}; the number of partitions of 3*(n+1)-C and 3*(n+1)+C are equal; example: n=8; 3*n+3=27; a(8)=4; (21,1), (22,1),(23,2), (24,2), (25,3), (26,3), (27,4), (28,3), (29,3), (30,2), (31,2),(32,1), (33,1). - Paul Weisenhorn, Jun 01 2009
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REFERENCES
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A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
M. Jeger, Einfuehrung in die Kombinatorik, Band 2, Klett, 1975, pages 110- [From Paul Weisenhorn, Jun 01 2009]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1,-1,-1,0,2,2,0,-1,-1,-1,-1,2,1,-1).
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FORMULA
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a(n) is the coefficient of x^(3*n+3) from the g.f. Product_{s=1..6} (x^s-x^(n+1))/(1-x^s). - Paul Weisenhorn, Jun 01 2009
G.f.: (x^10+x^8+3*x^7+4*x^6+4*x^5+4*x^4+3*x^3+x^2+1) / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x+1)^3 *(x-1)^6). - Alois P. Heinz, Jul 26 2015
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MATHEMATICA
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LinearRecurrence[{1, 2, -1, -1, -1, -1, 0, 2, 2, 0, -1, -1, -1, -1, 2, 1, -1}, {1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137}, 100] (* Jean-François Alcover, Feb 25 2020 *)
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PROG
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(PARI) f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)); n=200; p=subst(subst(f, x, x+x*O(x^n)), z, z+z*O(z^n)); for(n=0, 60, print1(polcoeff(polcoeff(p, 3*n), n)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
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STATUS
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approved
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