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

0,3

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).

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

a(n) = floor((10*n^9 +2475*n^8 +260700*n^7 +15246000*n^6 +541340184*n^5 +11985724800*n^4 +163531843200*n^3 +1309872384000*n^2 +5575652433792*n)/13168189440000 - (n mod 2)*(n^4+110*n^3+4180*n^2+63525*n)/1474560 - ((n mod 3)*(n^2+55*n) + ((2*n^2+2*n) mod 3)*7*n)/8748 + ((n^2+3*n+2) mod 4)*n/1024 + (((n+4) mod 5)-(n mod 5))*n/1250 + 1). - Hoang Xuan Thanh, Apr 06 2026

Mathematica

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

Prog

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

Crossrefs

Essentially same as A026816.

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

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

Sequence in context: A242696 A218510 A026816 * A341914 A008633 A347576

Adjacent sequences: A008636 A008637 A008638 * A008640 A008641 A008642

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved