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%I #32 Jun 05 2021 08:43:18
%S 1,1,1,2,3,4,6,9,13,17,25,33,45,61,82,106,142,183,238,306,395,499,638,
%T 804,1014,1268,1586,1967,2447,3018,3721,4566,5598,6827,8328,10108,
%U 12257,14812,17884,21508,25856,30980,37076,44261,52776,62768,74578,88407,104681,123703,146018,172019,202445,237830,279087,326991,382706
%N Number of partitions of n the largest part of which, call it m, appears once, m-1 appears at most twice, m-2 appears at most thrice, etc.
%H Alois P. Heinz, <a href="/A244393/b244393.txt">Table of n, a(n) for n = 0..1000</a>
%e For n=6 the partitions counted are 6, 51, 42, 411, 321, 3111
%e The a(9) = 17 such partitions of 9 are:
%e 01: [ 3 2 2 1 1 ]
%e 02: [ 4 2 1 1 1 ]
%e 03: [ 4 2 2 1 ]
%e 04: [ 4 3 1 1 ]
%e 05: [ 4 3 2 ]
%e 06: [ 5 1 1 1 1 ]
%e 07: [ 5 2 1 1 ]
%e 08: [ 5 2 2 ]
%e 09: [ 5 3 1 ]
%e 10: [ 5 4 ]
%e 11: [ 6 1 1 1 ]
%e 12: [ 6 2 1 ]
%e 13: [ 6 3 ]
%e 14: [ 7 1 1 ]
%e 15: [ 7 2 ]
%e 16: [ 8 1 ]
%e 17: [ 9 ]
%p b:= proc(n, i, t) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1, `if`(t=1, 1, t+1))+add(
%p b(n-i*j, i-1, t+1), j=1..min(t, n/i))))
%p end:
%p a:= n-> b(n$2, 1):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jul 29 2017
%t nend=20;
%t For[n=1,n<=nend,n++,
%t count[n]=0;
%t Ip=IntegerPartitions[n];
%t For[i=1,i<=Length[Ip],i++,
%t m=Max[Ip[[i]]];
%t condition=True;
%t Tip=Tally[Ip[[i]]];
%t For[j=1,j<=Length[Tip],j++,
%t condition=condition&&(Tip[[j]][[2]]<= m-Tip[[j]][[1]]+1)];
%t If[condition,count[n]++(*;Print[Ip[[i]]]*)]];
%t ]
%t Table[count[i],{i,1,nend}]
%t (* Second program: *)
%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0,
%t b[n, i - 1, If[t == 1, 1, t + 1]] + Sum[
%t b[n - i*j, i - 1, t + 1], {j, 1, Min[t, n/i]}]]];
%t a[n_] := b[n, n, 1];
%t a /@ Range[0, 60] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ *)
%Y Cf. A244395.
%K nonn
%O 0,4
%A _David S. Newman_, Jul 03 2014
%E More terms from _Joerg Arndt_, Jul 03 2014
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