A008637 Number of partitions of n into at most 8 parts.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Cf. A008284.

Strictly different from A008631, although they have similar descriptions.

Programs

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 ]

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 */

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

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)

Extensions

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