A115584 Number of partitions of n in which each part k occurs more than k times.
1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 7, 7, 8, 8, 12, 9, 15, 14, 17, 18, 24, 21, 29, 29, 35, 35, 46, 42, 56, 54, 65, 67, 81, 77, 98, 95, 115, 114, 139, 135, 164, 165, 190, 195, 230, 225, 272, 271, 313, 321, 370, 374, 433, 441, 501, 514, 589, 592, 681, 698, 778, 809, 907
Offset: 0
Examples
a(2) = 1 because we have [1,1]; a(10) = 4 because we have [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1] and [1^10]. From _Gus Wiseman_, Apr 02 2019: (Start) The initial terms count the following integer partitions: 0: () 2: (11) 3: (111) 4: (1111) 5: (11111) 6: (222) 6: (111111) 7: (1111111) 8: (2222) 8: (22211) 8: (11111111) 9: (222111) 9: (111111111) 10: (22222) 10: (222211) 10: (2221111) 10: (1111111111) 11: (2222111) 11: (22211111) 11: (11111111111) 12: (3333) 12: (222222) 12: (2222211) 12: (22221111) 12: (222111111) 12: (111111111111) (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
Programs
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Maple
g:=product((1-x^k+x^(k*(k+1)))/(1-x^k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Mar 12 2006 # second Maple program: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1) +add(b(n-i*j, i-1), j=i+1..n/i))) end: a:= n-> b(n$2): seq(a(n), n=0..80); # Alois P. Heinz, Feb 09 2017
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Mathematica
CoefficientList[ Series[ Product[(1 - x^k + x^(k(k + 1)))/(1 - x^k), {k, 14}], {x, 0, 66}], x] (* Robert G. Wilson v, Mar 12 2006 *) Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)
Formula
G.f.: Product_{k>=1} (1-x^k+x^(k*(k+1)))/(1-x^k).
Extensions
More terms from Robert G. Wilson v and Emeric Deutsch, Mar 12 2006
Comments