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A001401 Number of partitions of n into at most 5 parts.
(Formerly M0642 N0237)
21
1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765, 4033, 4319 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

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

For n>4: also number of partitions of n into parts <= 5: a(n)=A026820(n,5). [From Reinhard Zumkeller, Jan 21 2010]

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

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=5 of Q(m,n) table.

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

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).

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.

LINKS

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

Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 354

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

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.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

B. Kisacanin, Mathematical Problems and Proofs, Plenum, New York, 1998, pp. 71-72.

Jon Perry, More Partition Function

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

FORMULA

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

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

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, Jun 27 2003

(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)).

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

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

a(n+5) = a(n) + A001400(n). - Ece Uslu, Esin Becenen, Jan 11 2016

EXAMPLE

(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.

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

MAPLE

with(combstruct):ZL6:=[S, {S=Set(Cycle(Z, card<6))}, unlabeled]:seq(count(ZL6, size=n), n=0..52); # Zerinvary Lajos, Sep 24 2007

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

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

MATHEMATICA

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)), {x, 0, 60} ], x ]

a[n_] := IntegerPartitions[n, 5] // Length; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Jul 13 2012 *)

PROG

(PARI) a(n)=#partitions(n, , 5) \\ Charles R Greathouse IV, Sep 15 2014

CROSSREFS

a(n) = A008284(n+5, 5), n >= 0.

Cf. A008619, A001400, A001399, A008667 (first differences).

First differences of A002622.

Sequence in context: A062684 A033485 A026811 * A008628 A038499 A118199

Adjacent sequences:  A001398 A001399 A001400 * A001402 A001403 A001404

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Michael Somos and Branislav Kisacanin (branislav.kisacanin(AT)delphiauto.com)

STATUS

approved

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Last modified June 21 11:49 EDT 2018. Contains 305619 sequences. (Running on oeis4.)