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