0,3

Number of partitions of n into at most 4 parts.

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].

Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes - Vladeta Jovovic (Vladeta(AT)Eunet.yu), Dec 27 1999

Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 12 2002

a(n) = coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.

Gerzson Keri and Patric R. J. Ostergard, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.

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).

T. D. Noe, Table of n, a(n) for n=0..1000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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

F. Ellermann, Illustration for A001400, A061924

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 353

Jon Perry, More Partition Functions

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.

Index entries for sequences related to linear recurrences with constant coefficients

a(n)=1+(a(n-2)+a(n-3)+a(n-4))-(a(n-5)+a(n-6)+a(n-7))+a(n-9) - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000

P(n, 4) = 1/288( 2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n)-32*pcr{1, -1, 0}(3, n)-36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).

Let c(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)), then a(n) = sum(i=0, floor(n/4), 1+ceil((n-4*i-1)/2)+c(n-4*i-3)). - Jon Perry (perry(AT)globalnet.co.uk), Jun 27 2003

Euler transform of finite sequence [1, 1, 1, 1].

(n choose 4)_q=(q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1))

(4 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.

A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;

with(combstruct):ZL5:=[S, {S=Set(Cycle(Z, card<5))}, unlabeled]:seq(count(ZL5, size=n), n=0..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2007

A001400:=-(-z**8+z**9+2*z**4-z**7-1-z)/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for an initial 1.]

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=4)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..55); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2009]

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]

(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; G:=MatrixGroup<4, K|q1, q2, h>; MolienSeries(G);

Essentially same as A026810. Partial sums of A005044. Cf. A070083.

a(n)=A008284(n+4, 4), n >= 0.

Cf. A008619, A001399, A001401, A117486.

First differences of A002621.

Sequence in context: A104738 A028309 A026810 this_sequence A008773 A008772 A008771

Adjacent sequences: A001397 A001398 A001399 this_sequence A001401 A001402 A001403

nonn

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 29 2000