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A008641 Number of partitions of n into at most 12 parts. 3
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849, 56297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

With a different offset, number of partitions of n in which the greatest part is 12.

Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [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 361

Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

FORMULA

G.f.: 1/Product_{k=1..12}(1-x^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - David Neil McGrath, Jul 28 2015

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)/(1-x^12)

with(combstruct):ZL13:=[S, {S=Set(Cycle(Z, card<13))}, unlabeled]:seq(count(ZL13, size=n), n=0..46); # Zerinvary Lajos, Sep 24 2007

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=12)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # Zerinvary Lajos, Mar 21 2009

MATHEMATICA

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]

Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]

CROSSREFS

a(n) = A008284(n+12, 12), n >= 0.

Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.

Sequence in context: A242697 A218512 A008635 * A242698 A242699 A194439

Adjacent sequences:  A008638 A008639 A008640 * A008642 A008643 A008644

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)