%I A001400 M0627 N0229
%S A001400 1,1,2,3,5,6,9,11,15,18,23,27,34,39,47,54,64,72,84,94,108,120,136,150,
%T A001400 169,185,206,225,249,270,297,321,351,378,411,441,478,511,551,588,632,
%U A001400 672,720,764,816,864,920,972,1033,1089,1154,1215,1285,1350,1425,1495
%N A001400 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%C A001400 Number of partitions of n into at most 4 parts.
%C A001400 Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, 
               Chap. 7].
%C A001400 Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes 
               - Vladeta Jovovic (Vladeta(AT)Eunet.yu), Dec 27 1999
%C A001400 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
%C A001400 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
%D A001400 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,
               n) table; p. 120, P(n,4).
%D A001400 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D A001400 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 275.
%D A001400 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.
%D A001400 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001400 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001400 T. D. Noe, <a href="b001400.txt">Table of n, a(n) for n=0..1000</a>
%H A001400 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/
               ~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, 
               Springer, Berlin, 2006.
%H A001400 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001400 F. Ellermann, <a href="a061924.txt">Illustration for A001400, A061924</
               a>
%H A001400 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=353">
               Encyclopedia of Combinatorial Structures 353</a>
%H A001400 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/
               morepartitionfunction.htm">More Partition Functions</a>
%H A001400 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001400 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001400 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A001400 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
%F A001400 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).
%F A001400 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
%F A001400 Euler transform of finite sequence [1, 1, 1, 1].
%F A001400 (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))
%e A001400 (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.
%p A001400 A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/
               144); fi;
%p A001400 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
%p A001400 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.]
%p A001400 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]
%t A001400 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 
               0, 65} ], x ]
%o A001400 (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);
%Y A001400 Essentially same as A026810. Partial sums of A005044. Cf. A070083.
%Y A001400 a(n)=A008284(n+4, 4), n >= 0.
%Y A001400 Cf. A008619, A001399, A001401, A117486.
%Y A001400 First differences of A002621.
%Y A001400 Sequence in context: A104738 A028309 A026810 this_sequence A008773 A008772 
               A008771
%Y A001400 Adjacent sequences: A001397 A001398 A001399 this_sequence A001401 A001402 
               A001403
%K A001400 nonn
%O A001400 0,3
%A A001400 N. J. A. Sloane (njas(AT)research.att.com).
%E A001400 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 29 2000