A000601 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).

1, 2, 4, 7, 11, 16, 23, 31, 41, 53, 67, 83, 102, 123, 147, 174, 204, 237, 274, 314, 358, 406, 458, 514, 575, 640, 710, 785, 865, 950, 1041, 1137, 1239, 1347, 1461, 1581, 1708, 1841, 1981, 2128, 2282, 2443, 2612, 2788, 2972, 3164, 3364, 3572, 3789, 4014

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Molien series for 4-dimensional representation of S_3 [Nebe, Rains, Sloane, Chap. 7].

From Thomas Wieder, Feb 11 2007: (Start)

If P(i,k) denotes the number of integer partitions of i into k parts and if k=3, then a(n) = Sum_{i=k..n+2} P(i,k). See also A002620 = Quarter-squares, this sequence follows for k=2 as pointed out by Rick L. Shepherd, Feb 27 2004.

For example, a(n=6)=16 because there are 16 integer partitions of n=3,4,...,n+2=8 with k=3 parts:

[[1, 1, 1]],

[[2, 1, 1]],

[[3, 1, 1], [2, 2, 1]]

[[4, 1, 1], [3, 2, 1], [2, 2, 2]],

[[5, 1, 1], [4, 2, 1], [3, 3, 1], [3, 2, 2]],

[[6, 1, 1], [5, 2, 1], [4, 3, 1], [4, 2, 2], [3, 3, 2]]. (End)

Let P(i,k) be the number of integer partitions of n into k parts. Then if k=3 we have a(n) = Sum_{i=k..n} P(i,k=3). - Thomas Wieder, Feb 20 2007

Number of equivalence classes of 3 X n binary matrices when one can permute rows, permute columns and complement columns. - Max Alekseyev, Feb 05 2010

Convolution of the sequences whose n-th terms are given by 1+[n/2] and 1+[n/3], where []=floor. - Clark Kimberling, May 28 2012

Number of partitions of n into two sorts of 1, and one sort each of 2 and 3. - Joerg Arndt, May 05 2014

a(n-3) is the number of partitions mu of 2n of length 4 such that mu has an even number of even entries and the transpose of mu has an even number of even entries (see below example). - John M. Campbell, Feb 03 2016

Number of partitions of 2n+8 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both odd. Also, number of partitions of 2n+4 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both even. - Wesley Ivan Hurt, Jan 19 2021

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 23*x^6 + 31*x^7 + ...
From _John M. Campbell_, Feb 03 2016:  (Start)
For example, letting n=6, there are a(n-3)=a(3)=7 partitions mu of 12 of length 4 such mu has an even number of even entries and the transpose of mu has an even number of even entries: (8,2,1,1), (6,4,1,1), (6,3,2,1), (6,2,2,2), (4,4,3,1), (4,4,2,2), (4,3,3,2). For example, the partition
   oooooo
   ooo
   oo
   o
has 2 even entries and the transpose
   oooo
   ooo
   oo
   o
   o
   o
has an even number of even entries. (End)
		

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.
• 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

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• 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]
• L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
• Florent de Dinechin, Matei Istoan, Guillaume Sergent, Kinga Illyes, Bogdan Popa and Nicolas Brunie, Arithmetic around the bit heap, HAL Id: ensl-00738412, 2012. - From _N. J. A. Sloane_, Dec 31 2012
• E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
• E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
• M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649 - _Max Alekseyev_, Feb 05 2010
• H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
• H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 196
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• 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
• Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1)
• Index entries for two-way infinite sequences

Crossrefs

Cf. A002620, A006148, A006383.

Programs

Magma

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; U:=MatrixGroup<4,K|q2,h>; G:=MatrixGroup<4,K|q1,q2,h>; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(U);

Maple

A000601:=1/(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=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=3..52) ; # Zerinvary Lajos, Feb 07 2008

Mathematica

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)), {x, 0, 49}], x] (* Jean-François Alcover, Jul 20 2011 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,4,7,11,16,23},50] (* Harvey P. Dale, Mar 17 2013 *)
a[ n_] := Quotient[ 2 n^3 + 21 n^2 + 66 n, 72] + 1; (* Michael Somos, May 28 2014 *)

PARI

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

PARI

{a(n) = (2*n^3 + 21*n^2 + 66*n) \ 72 + 1}; /* Michael Somos, May 28 2014 */

Formula

a(n) = n^3/36 +7*n^2/24 +11*n/12 +119/144 +(-1)^n/16 + A057078(n)/9. - R. J. Mathar, Mar 14 2011

a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=11, a(5)=16, a(6)=23, a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). - Harvey P. Dale, Mar 17 2013

It appears that a(n) = ((4*n^3+42*n^2+140*n+102+21*(1+(-1)^n))/8-6*floor((2*n+5+3*(-1)^n)/12))/18. - Luce ETIENNE, May 05 2014

Euler transform of length 3 sequence [ 2, 1, 1]. - Michael Somos, May 28 2014

a(-7 - n) = -a(n). - Michael Somos, May 28 2014

Extensions

More terms from James Sellers, Feb 06 2000