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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
STATUS
approved