OFFSET
0,3
COMMENTS
For n>7: also number of partitions of n into parts <= 8: a(n)=A026820(n,8). - 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
From Vladimír Modrák, Jul 30 2022: (Start)
a(n) = Sum_{i_1=0..floor(n/3)} Sum_{i_2=0..floor(n/4)} Sum_{i_3=0..floor(n/5)} Sum_{i_4=0..floor(n/6)} Sum_{i_5=0..floor(n/7)} Sum_{i_6=0..floor(n/8)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6))/2).
a(n) = Sum_{i_1=0..floor(n/4)} Sum_{i_2=0..floor(n/5)} Sum_{i_3=0..floor(n/6)} Sum_{i_4=0..floor(n/7)} Sum_{i_5=0..floor(n/8)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5))^2+4)/12). (End)
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Dec 11 2000
STATUS
approved