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%I M0642 N0237 #103 Jul 13 2022 14:34:00
%S 1,1,2,3,5,7,10,13,18,23,30,37,47,57,70,84,101,119,141,164,192,221,
%T 255,291,333,377,427,480,540,603,674,748,831,918,1014,1115,1226,1342,
%U 1469,1602,1747,1898,2062,2233,2418,2611,2818,3034,3266,3507,3765,4033,4319
%N Number of partitions of n into at most 5 parts.
%C 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 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
%C For n > 4: also number of partitions of n into parts <= 5: a(n) = A026820(n,5). - _Reinhard Zumkeller_, Jan 21 2010
%C Number of different distributions of n+15 identical balls in 5 boxes as x,y,z,p,q where 0 < x < y < z < p < q. - _Ece Uslu_ and Esin Becenen, Jan 11 2016 [i.e., a(n) is the number of partitions of n+15 into 5 distinct parts. - _R. J. Mathar_, Feb 28 2021]
%C Tengely and Ulas prove that a(n) is a square only for n=1 and 2027. - _Michel Marcus_, Feb 11 2021
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=5 of Q(m,n) table.
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001401/b001401.txt">Table of n, a(n) for n = 0..1000</a>
%H Philippe Deléham, <a href="/A033485/a033485.pdf">Letter to N. J. A. Sloane, Apr 20 1998</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=354">Encyclopedia of Combinatorial Structures 354</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Gerzson Keri and Patric R. J. Östergård, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Keri/keri6.html">The Number of Inequivalent (2R+3,7)R Optimal Covering Codes</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
%H Clark Kimberling and John E. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H 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 Jon Perry, <a href="https://web.archive.org/web/20060923015527/http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Function</a>
%H Szabolcs Tengely and Maciej Ulas, <a href="https://arxiv.org/abs/2102.05352">Equal values of certain partition functions via Diophantine equations</a>, arXiv:2102.05352 [math.NT], 2021.
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).
%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
%F 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 Let a1(n) = Sum_{i=0..floor(n/3)} (1 + ceiling((n-3*i-1)/2)), a2(n) = Sum_{i=0..floor(n/4)} (1 + ceiling((n-4*i-1)/2) + a1(n-4*i-3)), then a(n) = Sum_{i=0..floor(n/5)} (1 + ceiling((n-5*i-1)/2) + a1(n-5*i-3) + a2(n-5*i-4)). - _Jon Perry_, Jun 27 2003
%F (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)).
%F a(n) = round(((n+5)^4 + 10*((n+5)^3 + (n+5)^2) - 75*(n+5) - 45*(n+5)*(-1)^(n+5))/2880). - _Washington Bomfim_, Jul 03 2012
%F a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n+15). - _David Neil McGrath_, Sep 13 2014
%F a(n+5) = a(n) + A001400(n) = A001400(n)+A026811(n). - _Ece Uslu_, Esin Becenen, Jan 11 2016
%F From _Vladimír Modrák_, Jul 13 2022: (Start)
%F a(n) = Sum_{k=0..floor(n/5)} Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0, n + 1 - 3*i - 4*j - 5*k))/2).
%F a(n) = Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j))^2+4)/12). (End)
%e (5 choose 5)_q = 1;
%e (6 choose 5)_q = q^5 + q^4 + q^3 + q^2 + q + 1;
%e (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;
%e (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;
%e so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
%e a(3) = 3, i.e., {1,2,3,4,8}, {1,2,3,5,7}, {1,2,4,5,6}. Number of different distributions of 18 identical balls in 5 boxes as x,y,z,p,q where 0 < x < y < z < p < q. - _Ece Uslu_, Esin Becenen, Jan 11 2016
%p with(combstruct):ZL6:=[S,{S=Set(Cycle(Z,card<6))}, unlabeled]:seq(count(ZL6,size=n),n=0..52); # _Zerinvary Lajos_, Sep 24 2007
%p 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..60); # _Alois P. Heinz_, Jul 31 2008
%p B:=[S,{S = Set(Sequence(Z,1 <= card),card <=5)},unlabelled]: seq(combstruct[count](B, size=n), n=0..52); # _Zerinvary Lajos_, Mar 21 2009
%t CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)), {x, 0, 60} ], x ]
%t a[n_] := IntegerPartitions[n, 5] // Length; Table[a[n], {n, 0, 52}] (* _Jean-François Alcover_, Jul 13 2012 *)
%t LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,1,2,3,5,7,10,13,18,23,30,37,47,57,70},60] (* _Harvey P. Dale_, Jan 05 2019 *)
%o (PARI) a(n)=#partitions(n,,5) \\ _Charles R Greathouse IV_, Sep 15 2014
%Y a(n) = A008284(n+5, 5), n >= 0.
%Y Cf. A008619, A001400, A001399, A008667 (first differences).
%Y First differences of A002622.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_ and Branislav Kisacanin (branislav.kisacanin(AT)delphiauto.com)
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