A000601 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)). (Formerly M1043 N0392)

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

0,2

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

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)

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

Example

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)

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

Prog

(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);
(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 */

Crossrefs

Cf. A002620, A006148, A006383.

Sequence in context: A114805 A196722 A181120 * A062433 A317910 A065095

Adjacent sequences: A000598 A000599 A000600 * A000602 A000603 A000604

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 06 2000

Status

approved