A008636 Number of partitions of n into at most 7 parts.

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241, 13534, 14950, 16475, 18138

Offset: 0

Author

N. J. A. Sloane, Mar 15 1996

Keywords

• nonn
• easy

Comments

Also, the number of partitions of n into parts <= 7: a(n) = A026820(n, 7). - Reinhard Zumkeller, Jan 21 2010

Counts unordered closed walks of weight n on a single vertex graph with 7 loops of weights 1, 2, 3, 4, 5, 6 and 7. - David Neil McGrath, Apr 11 2015

Number of different distributions of n+28 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - Ece Uslu and Esin Becenen, Jan 11 2016

Examples

			There are 28 partitions of 9 into parts less than or equal to 7. These are (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 11 2015
a(3) = 3, i.e., {1,2,3,4,5,7,9}, {1,2,3,4,6,7,8}, {1,2,3,4,5,6,10}. Number of different distributions of 31 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - _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
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 356
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1).

Crossrefs

Cf. A008284, A026820.

Programs

Maple

with(combstruct):ZL8:=[S,{S=Set(Cycle(Z,card<8))}, unlabeled]: seq(count(ZL8,size=n),n=0..48); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=7)},unlabelled]: seq(combstruct[count](B, size=n), n=0..48); # Zerinvary Lajos, Mar 21 2009

Mathematica

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 7} ], {x, 0, 60} ], x ]

PARI

{a(n)=(2*n^6+168*n^5+5530*n^4+90160*n^3+754299*n^2+(2988020+44800*(1-n%3))*n+6654375+1575*(3*n^2+84*n+511)*(-1)^n)\7257600}; \\ Tani Akinari, May 27 2014

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

a(n) = A008284(n+7, 7), n >= 0.

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28). - David Neil McGrath, Apr 11 2015

a(n+7) = a(n) + A001402(n). - Ece Uslu, Esin Becenen, Jan 11 2016

a(n) = A026813(n+7). - R. J. Mathar, Feb 13 2019

From Vladimír Modrák, Jul 30 2022: (Start)

a(n) = Sum_{p=0..floor(n/7)} Sum_{m=0..floor(n/6)} 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 - 6*m - 7*p))/2).

a(n) = Sum_{m=0..floor(n/7)} Sum_{k=0..floor(n/6)} Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j - 6*k - 7*m))^2+4)/12). (End)

Extensions

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