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%I #31 Apr 18 2017 07:03:05
%S 1,1,2,3,5,7,11,15,22,30,42,56,77,100,133,172,224,285,366,460,582,725,
%T 905,1116,1380,1686,2063,2503,3036,3655,4401,5262,6290,7476,8877,
%U 10489,12384,14552,17084,19978,23334,27156,31570,36578,42333,48849,56297
%N Number of partitions of n into at most 12 parts.
%C With a different offset, number of partitions of n in which the greatest part is 12.
%C Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [_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="/A008641/b008641.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=361">Encyclopedia of Combinatorial Structures 361</a>
%H <a href="/index/Rec#order_78">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).
%F G.f.: 1/Product_{k=1..12}(1-x^k).
%F a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - _David Neil McGrath_, Jul 28 2015
%p 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)/(1-x^11)/(1-x^12)
%p with(combstruct):ZL13:=[S,{S=Set(Cycle(Z,card<13))}, unlabeled]:seq(count(ZL13,size=n),n=0..46); # _Zerinvary Lajos_, Sep 24 2007
%p B:=[S,{S = Set(Sequence(Z,1 <= card),card <=12)},unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # _Zerinvary Lajos_, Mar 21 2009
%t CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]
%t Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]
%Y a(n) = A008284(n+12, 12), n >= 0.
%Y Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Dec 11 2000
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