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 A183561 Number of partitions of n containing a clique of size 4. 12
 1, 0, 1, 1, 3, 3, 5, 6, 10, 13, 20, 23, 35, 44, 61, 78, 103, 131, 174, 219, 285, 355, 456, 567, 721, 894, 1117, 1382, 1718, 2109, 2607, 3180, 3902, 4747, 5789, 7010, 8500, 10251, 12373, 14867, 17868, 21369, 25584, 30505, 36372, 43233 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,5 COMMENTS All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique. LINKS Alois P. Heinz, Table of n, a(n) for n = 4..1000 FORMULA G.f.: (1-Product_{j>0} (1-x^(4*j)+x^(5*j))) / (Product_{j>0} (1-x^j)). EXAMPLE a(10) = 5, because 5 partitions of 10 contain (at least) one clique of size 4: [1,1,1,1,2,2,2], [1,1,2,2,2,2], [1,1,1,1,3,3], [1,1,1,1,2,4], [1,1,1,1,6]. MAPLE b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],       add((l->`if`(j=4, [l[1]\$2], l))(b(n-i*j, i-1)), j=0..n/i)))     end: a:= n-> (l-> l[2])(b(n, n)): seq(a(n), n=4..50); MATHEMATICA max = 50; f = (1 - Product[1 - x^(4j) + x^(5j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x] , 4] (* Jean-François Alcover, Oct 01 2014 *) CROSSREFS 4th column of A183568. Cf. A000041, A183558, A183559, A183560, A183562, A183563, A183564, A183565, A183566, A183567. Sequence in context: A241090 A091607 A276434 * A300183 A222704 A095950 Adjacent sequences:  A183558 A183559 A183560 * A183562 A183563 A183564 KEYWORD nonn AUTHOR Alois P. Heinz, Jan 05 2011 STATUS approved

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Last modified February 26 03:09 EST 2020. Contains 332272 sequences. (Running on oeis4.)