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%I A001401 M0642 N0237
%S A001401 1,1,2,3,5,7,10,13,18,23,30,37,47,57,70,84,101,119,141,164,192,221,255,
%T A001401 291,333,377,427,480,540,603,674,748,831,918,1014,1115,1226,1342,1469,
%U A001401 1602,1747,1898,2062,2233,2418,2611,2818,3034,3266,3507,3765,4033,4319
%N A001401 Number of partitions of n into at most 5 parts.
%C A001401 a(n) = T_{r}(n) for r large, where T_{r}(n) = number of outcomes in which 
               r indistinguishable dice yield a sum r+n-1.
%C A001401 a(n) = coefficient of q^n in the expansion of (m choose 5)_q as m goes 
               to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
%D A001401 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=5 of Q(m,
               n) table.
%D A001401 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D A001401 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 A001401 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001401 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001401 T. D. Noe, <a href="b001401.txt">Table of n, a(n) for n=0..1000</a>
%H A001401 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=354">
               Encyclopedia of Combinatorial Structures 354</a>
%H A001401 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Partial Complements and Transposable Dispersions</
               a>, J. Integer Seqs., Vol. 7, 2004.
%H A001401 B. Kisacanin, <a href="http://www.wkap.nl/book.htm/0-306-45967-1">Mathematical 
               Problems and Proofs</a>, Plenum, New York, 1998, pp. 71-72.
%H A001401 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/
               morepartitionfunction.htm">More Partition Function</a>
%F A001401 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
%F A001401 a(n)=1+(a(n-2)+a(n-3)+a(n-4))-(a(n-6)+(2*a(n-7))+a(n-8))+(a(n-10)+a(n-11)+a(n-12))-a(n-14) 
               - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
%F A001401 Let a1(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)), a2(n)=sum(i=0, floor(n/
               4), 1+ceil((n-4*i-1)/2)+a1(n-4*i-3)), then a(n)=sum(i=0, floor(n/
               5), 1+ceil((n-5*i-1)/2)+a1(n-5*i-3)+a2(n-5*i-4)). - Jon Perry (perry(AT)globalnet.co.uk), 
               Jun 27 2003
%F A001401 (n choose 5)_q=(q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)*(q^(n-4)-1)/
               ((q^5-1)*(q^4-1)*(q^3-1)*(q^2-1)*(q-1))
%e A001401 (5 choose 5)_q = 1, (6 choose 5)_q = q^5 + q^4 + q^3 + q^2 + q + 1, (7 
               choose 5)_q = q^10 + q^9 + 2*q^8 + 2*q^7 + 3*q^6 + 3*q^5 + 3*q^4 
               + 2*q^3 + 2*q^2 + q + 1, (8 choose 5)_q = q^15 + q^14 + 2*q^13 + 
               3*q^12 + 4*q^11 + 5*q^10 + 6*q^9 + 6*q^8 + 6*q^7 + 6*q^6 + 5*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 A001401 with(combstruct):ZL6:=[S,{S=Set(Cycle(Z,card<6))}, unlabeled]:seq(count(ZL6,
               size=n),n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 24 2007
%p A001401 (Maple) a := n -> (Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then 
               [1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1][i] else 0 fi)^n)[1,
               1]; seq (a(n), n=0..52); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Jul 31 2008]
%p A001401 B:=[S,{S = Set(Sequence(Z,1 <= card),card <=5)},unlabelled]: seq(combstruct[count](B, 
               size=n), n=0..52);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 21 2009]
%t A001401 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 
               - x^5)), {x, 0, 60} ], x ]
%Y A001401 a(n)=A008284(n+5, 5), n >= 0.
%Y A001401 Cf. A008619, A001400, A001399.
%Y A001401 First differences of A002622.
%Y A001401 Sequence in context: A062684 A033485 A026811 this_sequence A008628 A038499 
               A118199
%Y A001401 Adjacent sequences: A001398 A001399 A001400 this_sequence A001402 A001403 
               A001404
%K A001401 nonn,easy,nice
%O A001401 0,3
%A A001401 N. J. A. Sloane (njas(AT)research.att.com).
%E A001401 Additional comments from Michael Somos and Branislav Kisacanin (branislav.kisacanin(AT)delphiauto.com)

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