A008640 Number of partitions of n into at most 11 parts.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 131, 169, 219, 278, 355, 445, 560, 695, 863, 1060, 1303, 1586, 1930, 2331, 2812, 3370, 4035, 4802, 5708, 6751, 7972, 9373, 11004, 12866, 15021, 17475, 20298, 23501, 27169, 31316, 36043, 41373, 47420, 54218, 61903, 70515, 80215, 91058, 103226, 116792, 131970, 148847

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n>10: also number of partitions of n into parts <= 11: a(n)=A026820(n,11). [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 360
• Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, -1, -1, -1, -2, -1, -1, 0, -1, 2, 2, 2, 2, 1, 1, 0, -1, -1, -2, -2, -2, -2, 1, 0, 1, 1, 2, 1, 1, 1, 0, 0, -1, -2, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1).

Crossrefs

Differs from A008634 at 55th term.

a(n) = A008284(n+11, 11), n >= 0.

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)/(1-x^9)/(1-x^10)/(1-x^11)
with(combstruct):ZL12:=[S,{S=Set(Cycle(Z,card<12))},unlabeled]: seq(count(ZL12,size=n),n=0..44); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=11)},unlabelled]: seq(combstruct[count](B, size=n), n=0..44); # Zerinvary Lajos, Mar 21 2009

Mathematica

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 11} ], {x, 0, 60} ], x ]

Formula

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-14) + 2*a(n-15) + a(n-16) - a(n-19) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - a(n-26) + 2*a(n-27) + 2*a(n-28) + 2*a(n-29) + 2*a(n-30) + a(n-31) + a(n-32) - a(n-34) - a(n-35) - 2*a(n-36) - 2*a(n-37) - 2*a(n-38) - 2*a(n-39) + a(n-40) + a(n-42) + a(n-43) + 2*a(n-44) + a(n-45) + a(n-46) + a(n-47) - a(n-50) - 2*a(n-51) - a(n-52) + a(n-59) + a(n-61) - a(n-64) - a(n-65) + a(n-66). - David Neil McGrath, Jul 27 2015

G.f.: 1 / prod(k=1..11, 1 - x^k ). - Joerg Arndt, Aug 04 2015