cp's OEIS Frontend

A244393 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.

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