A008638 Number of partitions of n into at most 9 parts.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 393, 488, 598, 732, 887, 1076, 1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824, 10156, 11648, 13338, 15224, 17354, 19720, 22380, 25331, 28629, 32278

Offset

0,3

Comments

For n > 8: also number of partitions of n into parts <= 9: a(n) = A026820(n, 9). - Reinhard Zumkeller, Jan 21 2010

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 358

Index entries for related partition-counting sequences

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

Formula

G.f.: 1/Product_{k=1..9} (1 - q^k).

a(n) = floor((30*n^8 + 5400*n^7 + 405300*n^6 + 16443000*n^5 + 390533640*n^4 + 5486840100*n^3 + 43691213950*n^2 + 175052776500*n + 256697834389)/438939648000 + (n + 1)*(2*n^2 + 133*n + 2597)*(-1)^n/147456 + (-1)^n*((n + 1)*(n + 47)*(-1)^floor(n/3 + 2/3) + (2*n^2 + 90*n + 127)*(-1)^floor(n/3 + 1/3) + (n + 2)*(n + 40)*(-1)^floor(n/3))/17496 + 1/256*((-1)^((2*n + (-1)^n - 1)/4)*floor((n + 2)/2)) + 1/2). - Tani Akinari, Oct 20 2012

a(n) = a(n-9) + A008637(n). - Vladimír Modrák, Sep 28 2020

From Vladimír Modrák, Aug 09 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)} Sum_{i_7=0..floor(n/9)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6 - 9*i_7))/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)} Sum_{i_6=0..floor(n/9)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5 - 9*i_6))^2+4)/12). (End)

Mathematica

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 9} ], {x, 0, 60} ], x ]

Crossrefs

Essentially same as A026815.

a(n) = A008284(n+9, 9), n >= 0.

Cf. A288344 (partial sums), A266777 (first differences).

Sequence in context: A218509 A026815 A341913 * A008632 A347575 A238867

Adjacent sequences: A008635 A008636 A008637 * A008639 A008640 A008641

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved