A008639 Number of partitions of n into at most 10 parts.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 128, 164, 212, 267, 340, 423, 530, 653, 807, 984, 1204, 1455, 1761, 2112, 2534, 3015, 3590, 4242, 5013, 5888, 6912, 8070, 9418, 10936, 12690, 14663, 16928, 19466, 22367, 25608, 29292, 33401, 38047

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n > 9: also number of partitions of n into parts <= 10: a(n) = A026820(n, 10). - 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 359
• 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, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Crossrefs

Essentially same as A026816.

a(n) = A008284(n + 10, 10), n >= 0.

Cf. A266778 (first differences), A288345 (partial sums).

Programs

Mathematica

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 10} ], {x, 0, 60} ], x ]

PARI

Vec(1/prod(k=1,10,1-x^k)+O(x^99)) \\ Charles R Greathouse IV, May 06 2015

Formula

G.f.: 1/Product_{k=1..10} (1 - x^k). - David Neil McGrath, Apr 29 2015

a(n) = a(n-10) + A008638(n). - Vladimír Modrák, Sep 29 2020