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A008636 Number of partitions of n into at most 7 parts. 5
1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241, 13534, 14950, 16475, 18138 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, the number of partitions of n into parts <= 7: a(n) = A026820(n, 7). - Reinhard Zumkeller, Jan 21 2010

Counts unordered closed walks of weight n on a single vertex graph with 7 loops of weights 1, 2, 3, 4, 5, 6 and 7. - David Neil McGrath, Apr 11 2015

Number of different distributions of n+28 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - Ece Uslu and Esin Becenen, Jan 11 2016

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 356

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

FORMULA

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

a(n) = A008284(n+7, 7), n >= 0.

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28). - David Neil McGrath, Apr 11 2015

a(n+7) = a(n) + A001402(n). - Ece Uslu, Esin Becenen, Jan 11 2016

a(n) = A026813(n+7). - R. J. Mathar, Feb 13 2019

EXAMPLE

There are 28 partitions of 9 into parts less than or equal to 7. These are (72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - David Neil McGrath, Apr 11 2015

a(3) = 3, i.e., {1,2,3,4,5,7,9}, {1,2,3,4,6,7,8}, {1,2,3,4,5,6,10}. Number of different distributions of 31 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - Ece Uslu, Esin Becenen, Jan 11 2016

MAPLE

with(combstruct):ZL8:=[S, {S=Set(Cycle(Z, card<8))}, unlabeled]: seq(count(ZL8, size=n), n=0..48); # Zerinvary Lajos, Sep 24 2007

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=7)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..48); # Zerinvary Lajos, Mar 21 2009

MATHEMATICA

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 7} ], {x, 0, 60} ], x ]

PROG

(PARI) {a(n)=(2*n^6+168*n^5+5530*n^4+90160*n^3+754299*n^2+(2988020+44800*(1-n%3))*n+6654375+1575*(3*n^2+84*n+511)*(-1)^n)\7257600}; \\ Tani Akinari, May 27 2014

CROSSREFS

Cf. A008284, A026820.

Sequence in context: A218507 A339672 A026813 * A008630 A238865 A326978

Adjacent sequences:  A008633 A008634 A008635 * A008637 A008638 A008639

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mar 15 1996

EXTENSIONS

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

STATUS

approved

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Last modified April 13 12:29 EDT 2021. Contains 342936 sequences. (Running on oeis4.)