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A008639 Number of partitions of n into at most 10 parts. 4
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 (list; graph; refs; listen; history; text; internal format)
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

Counts unordered walks on a single vertex graph containing loops of weight (1,2,3,4,5,6,7,8,9,10). - David Neil McGrath, Apr 29 2015

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

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-11) + a(n-12) + a(n-13) + a(n-14) + 2*a(n-15) - a(n-18) - a(n-19) - a(n-20) - a(n-21) - 3*a(n-22) + a(n-25) + a(n-26) + 2*a(n-27) + 2*a(n-28) + a(n-29) + a(n-30) - 3*a(n-33) - a(n-34) - a(n-35) - a(n-36) - a(n-37) + 2*a(n-40) + a(n-41) + a(n-42) + a(n-43) - a(n-44) - a(n-48) - a(n-50) + a(n-53) + a(n-54) - a(n-55). - David Neil McGrath, Apr 28 2015

G.f.: 1/(prod[1-x^k] k=1...10). - David Neil McGrath, Apr 29 2015

EXAMPLE

a(10)=42. These are ([10]),(91),(82),(811),(73),(721),(7111),(64),(631),(622),(6211),(61111),(55),(541),(532),(5311),(5221),(52111),(511111),(442),(4411),(433),(4321),(43111),(4222),(42211),(421111),(4111111),(3331),(3322),(33211),(331111),(32221),(322111),(3211111),(31111111),(22222),(222211),(2221111),(22111111),(211111111),(1111111111). - David Neil McGrath, Apr 29 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)

with(combstruct):ZL11:=[S, {S=Set(Cycle(Z, card<11))}, unlabeled]:seq(count(ZL11, size=n), n=0..46); # Zerinvary Lajos, Sep 24 2007

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=10)}, 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, 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 * A008633 A238868 A035999

Adjacent sequences:  A008636 A008637 A008638 * A008640 A008641 A008642

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 17 23:26 EST 2019. Contains 329242 sequences. (Running on oeis4.)