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 A258289 Number of partitions of 1, 2, 3, or more copies of n into distinct parts. 1
 1, 1, 1, 3, 3, 7, 9, 17, 21, 43, 57, 109, 157, 301, 447, 895, 1307, 2663, 4207, 8463, 13283, 28489, 45151, 95485, 157767, 336711, 561603, 1236963, 2061173, 4567227, 7946575, 17516101, 30324977, 69519697, 121465499, 276609723, 496333307, 1137900605 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..37. FORMULA a(n) = Sum_{k=1..A065033(n)} A258280(n,k). a(n) = Sum_{k=1..max(1,ceiling(n/2))} 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j). EXAMPLE a(0) = 1: []. a(1) = 1: [1]. a(2) = 1: [2]. a(3) = 3: [3], [2,1], [3;2,1]. a(4) = 3: [4], [3,1], [4;3,1]. a(5) = 7: [5], [4,1], [3,2], [5;4,1], [5;3,2], [4,1;3,2], [5;4,1;3,2]. a(7) = 17: [7], [6,1], [5,2], [4,3], [4,2,1], [7;6,1], [7;5,2], [7;4,3], [7;4,2,1], [6,1;5,2], [6,1;4,3], [5,2;4,3], [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3], [7;6,1;5,2;4,3]. MAPLE b:= proc() option remember; local m; m:= args[nargs]; `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] \$t=2..nargs), `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0, b(args[t] \$t=1..nargs-1, m-1)+add(`if`(args[j]-m<0, 0, b(sort([seq(args[i]-`if`(i=j, m, 0), i=1..nargs-1)])[] , m-1)), j=1..nargs-1)))) end: a:= n-> add(b(n\$k+1)/k!, k=1..max(1, ceil(n/2))): seq(a(n), n=0..20); MATHEMATICA disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&]; T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length; a[n_] := a[n] = If[n == 0, 1, Sum[T[n, k], {k, 1, Quotient[n+1, 2]}]]; Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, May 01 2022 *) CROSSREFS Cf. A000009, A065033, A258280. Sequence in context: A237666 A285187 A034411 * A066983 A048240 A122012 Adjacent sequences: A258286 A258287 A258288 * A258290 A258291 A258292 KEYWORD nonn AUTHOR Alois P. Heinz, May 25 2015 STATUS approved

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Last modified September 8 01:34 EDT 2024. Contains 375749 sequences. (Running on oeis4.)