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 A001977 Number of partitions of 3n into n parts from the set {0, 1, ..., 6} (repetitions admissible). (Formerly M3335 N1342) 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 REFERENCES 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 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. 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. [Annotated scanned copy] Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019. 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). FORMULA 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 MATHEMATICA 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 *) PROG (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 CROSSREFS Sequence in context: A195334 A009918 A008085 * A008373 A008374 A201686 Adjacent sequences:  A001974 A001975 A001976 * A001978 A001979 A001980 KEYWORD nonn,easy AUTHOR EXTENSIONS Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008 STATUS approved

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