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A008763
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G.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).
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14
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0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
Coefficient of s(2n-8) in s(n-4,n-4) * s(n-4,n-4) * s(n-4,n-4) * s(n-4,n-4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Dec 22 2005
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REFERENCES
| G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2.
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
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FORMULA
| Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
(1/144) {2n^3 + 9n[(-1)^n - 1] - 16[[n is 2 mod 3]-[n is 1 mod 3]]}.
a(n)=(1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*pi*n/3) [From Richard Choulet (richardchoulet(AT)yahoo.fr), Nov 27 2008]
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EXAMPLE
| a(7) = 4:
41 32 31 22
11 11 21 21
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MAPLE
| (Maple) a := n -> (Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 2, -1, -2, -1, 2, 1, -1][i] else 0 fi)^n)[1, 5]; seq (a(n), n=0..56); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 31 2008]
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MATHEMATICA
| CoefficientList[Series[x^4 / ((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 56}], x] (* From Jean-François Alcover, Mar 30 2011 *)
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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; H:=MatrixGroup<4, K|q1, q2, h, p1>; MolienSeries(H);
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CROSSREFS
| Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.
Sequence in context: A103054 A140208 A098390 * A005896 A147953 A163468
Adjacent sequences: A008760 A008761 A008762 * A008764 A008765 A008766
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Stephen P Humphries (steve(AT)math.byu.edu)
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EXTENSIONS
| Entry revised Dec 25 2003
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