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A008639 Number of partitions of n into at most 10 parts.

%I A008639 #85 Sep 11 2022 09:35:05
%S A008639 1,1,2,3,5,7,11,15,22,30,42,55,75,97,128,164,212,267,340,423,530,653,
%T A008639 807,984,1204,1455,1761,2112,2534,3015,3590,4242,5013,5888,6912,8070,
%U A008639 9418,10936,12690,14663,16928,19466,22367,25608,29292,33401,38047
%N A008639 Number of partitions of n into at most 10 parts.
%C A008639 For n > 9: also number of partitions of n into parts <= 10: a(n) = A026820(n, 10). - _Reinhard Zumkeller_, Jan 21 2010
%D A008639 A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
%D A008639 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%H A008639 T. D. Noe, <a href="/A008639/b008639.txt">Table of n, a(n) for n = 0..1000</a>
%H A008639 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=359">Encyclopedia of Combinatorial Structures 359</a>
%H A008639 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A008639 <a href="/index/Rec#order_55">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A008639 G.f.: 1/Product_{k=1..10} (1 - x^k). - _David Neil McGrath_, Apr 29 2015
%F A008639 a(n) = a(n-10) + A008638(n). - _Vladimír Modrák_, Sep 29 2020
%t A008639 CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 10} ], {x, 0, 60} ], x ]
%o A008639 (PARI) Vec(1/prod(k=1,10,1-x^k)+O(x^99)) \\ _Charles R Greathouse IV_, May 06 2015
%Y A008639 Essentially same as A026816.
%Y A008639 a(n) = A008284(n + 10, 10), n >= 0.
%Y A008639 Cf. A266778 (first differences), A288345 (partial sums).
%K A008639 nonn,easy
%O A008639 0,3
%A A008639 _N. J. A. Sloane_