A001973 Expansion of (1+x^3)/((1-x)*(1-x^2)^2*(1-x^3)). (Formerly M2441 N0969)

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

0,3

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)

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

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<r), U=Sequence(Z, 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 *)

Prog

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

Crossrefs

Sequence in context: A265063 A098202 A164653 * A248374 A020745 A232896

Adjacent sequences: A001970 A001971 A001972 * A001974 A001975 A001976

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved