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%I #73 Aug 09 2022 15:30:11
%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
%F From _Vladimír Modrák_, Aug 09 2022: (Start)
%F a(n) = Sum_{i_1=0..floor(n/3)} Sum_{i_2=0..floor(n/4)} Sum_{i_3=0..floor(n/5)} Sum_{i_4=0..floor(n/6)} Sum_{i_5=0..floor(n/7)} Sum_{i_6=0..floor(n/8)} Sum_{i_7=0..floor(n/9)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6 - 9*i_7))/2).
%F a(n) = Sum_{i_1=0..floor(n/4)} Sum_{i_2=0..floor(n/5)} Sum_{i_3=0..floor(n/6)} Sum_{i_4=0..floor(n/7)} Sum_{i_5=0..floor(n/8)} Sum_{i_6=0..floor(n/9)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5 - 9*i_6))^2+4)/12). (End)
%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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