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A008637 Number of partitions of n into at most 8 parts. 5
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749, 11018, 12450, 14012, 15765, 17674, 19805, 22122 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n>7: also number of partitions of n into parts <= 8: a(n)=A026820(n,8). [From Reinhard Zumkeller, Jan 21 2010]

Molien series for finite Coxeter group of type A_8.

Number of different distributions of n+36 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0<x<y<z<p<q<m<n<h. - Ece Uslu and Esin Becenen, Jan 11 2016

REFERENCES

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.

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

LINKS

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

Tani Akinari, Formula for a(n)

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 357

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1).

FORMULA

G.f.: 1/((1-t)*(1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)). - N. J. A. Sloane, Jan 09 2016

a(n) = A008284(n+8, 8), n >= 0.

a(n) = floor((-1)^n*((n+1)*(-1)^(floor((n+2)/3))+(2*n+3)*(-1)^(floor((n+1)/3))+(n+2)*(-1)^(floor(n/3)))/972+(n+2)*((-1)^n+1)*(-1)^(n/2)/512+(n+18)*(6*n^6+648*n^5+27018*n^4+545616*n^3+5481213*n^2+25163028*n+39226571)/1219276800+(n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2). (See link.) - Tani Akinari, Oct 26 2012

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-9) + a(n-11) + 2*a(n-12) + a(n-13) + a(n-15) - a(n-16) - a(n-17) - 2*a(n-18) - a(n-19) - a(n-20) + a(n-21) + a(n-23) + 2*a(n-24) + a(n-25) - a(n-27) - a(n-29) - a(n-31) + a(n-34) + a(n-35) - a(n-36). - David Neil McGrath, Apr 14 2015

a(n+8) = a(n) + A008636(n). - Ece Uslu, Esin Becenen, Jan 11 2016

EXAMPLE

There are a(9)=29 partitions of 9 into parts less than or equal to 8. These are (81)(72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - David Neil McGrath, Apr 14 2015

a(3) = 3 i.e. {1,2,3,4,5,7,8,9},{1,2,3,4,5,6,8,10},{1,2,3,4,5,6,7,11} Number of different distributions of 39 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0<x<y<z<p<q<m<n<h. - Ece Uslu, Esin Becenen, Jan 11 2016

MAPLE

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)

with(combstruct):ZL9:=[S, {S=Set(Cycle(Z, card<9))}, unlabeled]:seq(count(ZL9, size=n), n=0..47); # Zerinvary Lajos, Sep 24 2007

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=8)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..47); # Zerinvary Lajos, Mar 21 2009

MATHEMATICA

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 8} ], {x, 0, 60} ], x ]

PROG

(Maxima)a(n):=floor((-1)^n*((n+1)*(-1)^floor((n+2)/3)+(2*n+3)*(-1)^floor((n+1)/3)+(n+2)*(-1)^floor(n/3))/972+(n+2)*((-1)^n+1)*(-1)^(n/2)/512+(n+18)*(6*n^6+648*n^5+27018*n^4+545616*n^3+5481213*n^2+25163028*n+39226571)/1219276800+(n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2); \\ Tani Akinari, Oct 25 2012

CROSSREFS

Cf. A008284.

Strictly different from A008631, although they have similar descriptions.

Sequence in context: A218508 A340719 A026814 * A008631 A238866 A035978

Adjacent sequences:  A008634 A008635 A008636 * A008638 A008639 A008640

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Robert G. Wilson v, Dec 11 2000

STATUS

approved

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Last modified May 17 12:55 EDT 2021. Contains 343971 sequences. (Running on oeis4.)