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A115584 Number of partitions of n in which each part k occurs more than k times.

%I A115584 #20 Mar 05 2024 12:28:02
%S A115584 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,
%T A115584 35,46,42,56,54,65,67,81,77,98,95,115,114,139,135,164,165,190,195,230,
%U A115584 225,272,271,313,321,370,374,433,441,501,514,589,592,681,698,778,809,907
%N A115584 Number of partitions of n in which each part k occurs more than k times.
%C A115584 The Heinz numbers of these partitions are given by A325127. - _Gus Wiseman_, Apr 02 2019
%H A115584 Alois P. Heinz, <a href="/A115584/b115584.txt">Table of n, a(n) for n = 0..10000</a>
%F A115584 G.f.: Product_{k>=1} (1-x^k+x^(k*(k+1)))/(1-x^k).
%e A115584 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].
%e A115584 From _Gus Wiseman_, Apr 02 2019: (Start)
%e A115584 The initial terms count the following integer partitions:
%e A115584    0: ()
%e A115584    2: (11)
%e A115584    3: (111)
%e A115584    4: (1111)
%e A115584    5: (11111)
%e A115584    6: (222)
%e A115584    6: (111111)
%e A115584    7: (1111111)
%e A115584    8: (2222)
%e A115584    8: (22211)
%e A115584    8: (11111111)
%e A115584    9: (222111)
%e A115584    9: (111111111)
%e A115584   10: (22222)
%e A115584   10: (222211)
%e A115584   10: (2221111)
%e A115584   10: (1111111111)
%e A115584   11: (2222111)
%e A115584   11: (22211111)
%e A115584   11: (11111111111)
%e A115584   12: (3333)
%e A115584   12: (222222)
%e A115584   12: (2222211)
%e A115584   12: (22221111)
%e A115584   12: (222111111)
%e A115584   12: (111111111111)
%e A115584 (End)
%p A115584 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
%p A115584 # second Maple program:
%p A115584 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A115584       b(n, i-1) +add(b(n-i*j, i-1), j=i+1..n/i)))
%p A115584     end:
%p A115584 a:= n-> b(n$2):
%p A115584 seq(a(n), n=0..80);  # _Alois P. Heinz_, Feb 09 2017
%t A115584 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 *)
%t A115584 Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>i,{i,Union[#]}]&]],{n,0,30}] (* _Gus Wiseman_, Apr 02 2019 *)
%Y A115584 Cf. A052335, A087153, A114639, A117144, A276429, A324572, A325127.
%K A115584 easy,nonn
%O A115584 0,7
%A A115584 _Vladeta Jovovic_, Mar 09 2006
%E A115584 More terms from _Robert G. Wilson v_ and _Emeric Deutsch_, Mar 12 2006