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A008638 Number of partitions of n into at most 9 parts. 5

%I

%S 1,1,2,3,5,7,11,15,22,30,41,54,73,94,123,157,201,252,318,393,488,598,

%T 732,887,1076,1291,1549,1845,2194,2592,3060,3589,4206,4904,5708,6615,

%U 7657,8824,10156,11648,13338,15224,17354,19720,22380,25331,28629,32278

%N Number of partitions of n into at most 9 parts.

%C For n > 8: also number of partitions of n into parts <= 9: a(n) = A026820(n, 9). - _Reinhard Zumkeller_, Jan 21 2010

%D A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.

%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

%H T. D. Noe, <a href="/A008638/b008638.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=358">Encyclopedia of Combinatorial Structures 358</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%H <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1,0,-1,0,0,-1,0,2,1,1,1,0, -1,-1,-1,-2,-1,-1,1,1,2,1,1,1,0, -1,-1,-1,-2,0,1,0,0,1,0,1,0,0,-1,-1,1)

%F G.f.: 1/Product_{k=1..9} (1 - q^k).

%F a(n) = floor((30*n^8 + 5400*n^7 + 405300*n^6 + 16443000*n^5 + 390533640*n^4 + 5486840100*n^3 + 43691213950*n^2 + 175052776500*n + 256697834389)/438939648000 + (n + 1)*(2*n^2 + 133*n + 2597)*(-1)^n/147456 + (-1)^n*((n + 1)*(n + 47)*(-1)^floor(n/3 + 2/3) + (2*n^2 + 90*n + 127)*(-1)^floor(n/3 + 1/3) + (n + 2)*(n + 40)*(-1)^floor(n/3))/17496 + 1/256*((-1)^((2*n + (-1)^n - 1)/4)*floor((n + 2)/2)) + 1/2). - _Tani Akinari_, Oct 20 2012

%F a(n) = a(n-9) + A008637(n). - _Vladimír Modrák_, Sep 28 2020

%t CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 9} ], {x, 0, 60} ], x ]

%Y Essentially same as A026815.

%Y a(n) = A008284(n+9, 9), n >= 0.

%Y Cf. A288344 (partial sums), A266777 (first differences).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified June 14 15:15 EDT 2021. Contains 345025 sequences. (Running on oeis4.)