0,2

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

Comments from Thomas Wieder (thomas.wieder(AT)t-online.de), 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 Shepherd (rshepherd2(AT)hotmail.com), 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 (thomas.wieder(AT)t-online.de), Feb 20 2007

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.

H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.

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).

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

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.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

A000601:=1/(z+1)/(z**2+z+1)/(z-1)**4; [S. 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 (zerinvarylajos(AT)yahoo.com), Feb 07 2008

(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);

Cf. A002620.

Sequence in context: A131075 A133523 A114805 this_sequence A062433 A065095 A005253

Adjacent sequences: A000598 A000599 A000600 this_sequence A000602 A000603 A000604

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 06 2000