A008763 - OEIS

%I #94 Jan 24 2025 18:36:28

%S 0,0,0,0,1,1,3,4,7,9,14,17,24,29,38,45,57,66,81,93,111,126,148,166,

%T 192,214,244,270,305,335,375,410,455,495,546,591,648,699,762,819,889,

%U 952,1029,1099,1183,1260,1352,1436,1536,1628,1736,1836,1953,2061,2187,2304,2439

%N Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

%C Number of 2 X 2 square partitions of n.

%C 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].

%C 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

%C pq

%C rs

%C with p >= q, p >= r, q >= s, r >= s.

%C The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(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_, Dec 22 2005

%C Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - _Michael Somos_, Jan 21 2015

%C Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - _Wesley Ivan Hurt_, Jan 19 2021

%D G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.

%D 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).

%D 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.

%H T. D. Noe, <a href="/A008763/b008763.txt">Table of n, a(n) for n = 0..1000</a>

%H Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, <a href="http://ftp.math.uni-rostock.de/pub/romako/heft68/bouroubi68.html">Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon</a>, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.

%H W. Duke, <a href="http://dx.doi.org/10.1155/S1073792893000121">On codes and Siegel modular forms</a>, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]

%H Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, <a href="https://arxiv.org/abs/2501.10267">Enumeration of partitions via socle reduction</a>, arXiv:2501.10267 [math.CO], 2025. See p. 40.

%H S. P. Humphries, <a href="http://www.math.byu.edu/~steve/">Home page</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=450">Encyclopedia of Combinatorial Structures 450</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=232">Encyclopedia of Combinatorial Structures 232</a>

%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/step2.txt">In the Elliptic Realm</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,-1,2,1,-1).

%F 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).

%F a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).

%F 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). - _Richard Choulet_, Nov 27 2008

%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - _Harvey P. Dale_, Mar 04 2012

%F a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012

%F a(n+1) - a(n) = A008731(n-3). - _R. J. Mathar_, Aug 06 2013

%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jan 21 2015

%F Euler transform of length 3 sequence [1, 2, 1]. - _Michael Somos_, Jun 26 2017

%e a(7) = 4:

%e 41 32 31 22

%e 11 11 21 21

%e G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...

%e a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015

%e a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015

%e a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

%p 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..60); # _Alois P. Heinz_, Jul 31 2008

%t CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* _Jean-François Alcover_, Mar 30 2011 *)

%t LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* _Harvey P. Dale_, Mar 04 2012 *)

%t a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* _Michael Somos_, Jan 21 2015 *)

%t a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* _Michael Somos_, Jan 21 2015 *)

%o (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);

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // _G. C. Greubel_, Sep 10 2019

%o (PARI) {a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* _Michael Somos_, Jan 21 2015 */

%o (PARI) a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ _Charles R Greathouse IV_, Feb 06 2017

%o (Sage)

%o def AA008763_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()

%o AA008763_list(60) # _G. C. Greubel_, Sep 10 2019

%o (GAP) a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # _G. C. Greubel_, Sep 10 2019

%Y See A266769 for a version without the four leading zeros.

%Y Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.

%Y First differences of A097701.

%Y Cf. A082424, A082437.

%K nonn,nice,easy

%O 0,7

%A _N. J. A. Sloane_, _Simon Plouffe_, _Stephen P. Humphries_

%E Entry revised Dec 25 2003