A001973 Expansion of (1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)).

1, 1, 3, 5, 8, 12, 18, 24, 33, 43, 55, 69, 86, 104, 126, 150, 177, 207, 241, 277, 318, 362, 410, 462, 519, 579, 645, 715, 790, 870, 956, 1046, 1143, 1245, 1353, 1467, 1588, 1714, 1848, 1988

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(1..3)=0; a(n) is the number of partitions of 2*(n+1) with 4 different numbers from the set {1,...,n}; the number of partitions of 2*n + 2 - C and 2*n + 2 + C are equal; example: n=6; 2*n + 2 = 14; a(6)=3; (10,1), (11,1), (12,2), (13,2), (14,3), (15,2), (16,2), (17,1), (18,1). - Paul Weisenhorn, Jun 01 2009. [I believe this comment refers to the sequence 0, 0, 0, 1, 1, 3, 5, ... with offset 1. - N. J. A. Sloane, Mar 30 2023]

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, Klett, 1975, pages 110ff. [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

• Vincenzo Librandi, 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. [Annotated scanned copy]
• Shalosh B. Ekhad, 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.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1)

Programs

Maple

A001973:=(1-z+z**2)/(z+1)/(z**2+z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=2)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=6..45) ; # Zerinvary Lajos, Feb 07 2008

Mathematica

CoefficientList[Series[(1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)

PARI

Vec((1+x^3)/((1-x)*(1-x^2)^2*(1-x^3))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) is the coefficient of x^(2*n+2) in the g.f. Product_{s=1..4} (x^s - x^(n+1))/(1-x^s). - Paul Weisenhorn, Jun 01 2009

a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). Vincenzo Librandi, Jun 11 2012