A320612
Number of parts in all partitions of n in which no part occurs more than nine times.
Original entry on oeis.org
1, 3, 6, 12, 20, 35, 54, 86, 128, 182, 264, 376, 520, 718, 978, 1318, 1761, 2338, 3070, 4008, 5206, 6707, 8604, 10982, 13933, 17604, 22155, 27745, 34627, 43061, 53338, 65859, 81074, 99458, 121687, 148469, 180633, 219202, 265386, 320473, 386147, 464245, 556925
Offset: 1
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b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(9*i*(i+1)/2 [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 9))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);
-
Table[Length[ Flatten[Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]] <= 9 &]]], {n, 43}] (* Robert Price, Jul 31 2020 *)
A185350
Number of parts in all partitions of n in which no part occurs more than twice.
Original entry on oeis.org
0, 1, 3, 3, 8, 11, 17, 23, 36, 48, 69, 88, 125, 157, 212, 271, 356, 445, 574, 711, 906, 1118, 1400, 1711, 2125, 2583, 3171, 3828, 4666, 5604, 6777, 8095, 9730, 11567, 13815, 16357, 19429, 22910, 27077, 31801, 37432, 43802, 51338, 59871, 69914, 81273, 94562
Offset: 0
a(6) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
a(7) = 23: [7], [6,1], [5,2], [4,3], [5,1,1], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1].
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b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 2))))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);
-
b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]] j}, {j, 0, Min[n/i, k]} ] ] ];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n],Max[Length/@Split[#]]<3&]]],{n,0,50}] (* Harvey P. Dale, Jul 04 2023 *)
A320613
Number of parts in all partitions of n in which no part occurs more than ten times.
Original entry on oeis.org
1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 264, 387, 531, 741, 1001, 1366, 1809, 2425, 3170, 4172, 5398, 6997, 8948, 11482, 14547, 18451, 23193, 29157, 36355, 45347, 56150, 69509, 85565, 105216, 128732, 157393, 191545, 232870, 282033, 341164, 411251, 495197, 594369
Offset: 1
-
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(5*i*(i+1) [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 10))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);
-
Table[Length[Flatten[Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]] <= 10 &]]], {n, 43}] (* Robert Price, Jul 31 2020 *)
A364245
Number of parts in all partitions of 2n into parts with multiplicity <= n.
Original entry on oeis.org
0, 1, 8, 24, 65, 150, 330, 657, 1274, 2338, 4172, 7203, 12171, 20045, 32474, 51623, 80867, 124841, 190406, 286857, 427758, 631367, 923544, 1339226, 1926798, 2751094, 3900931, 5494411, 7690923, 10701618, 14808183, 20380969, 27910066, 38035633, 51597166, 69685656
Offset: 0
a(2) = 8 = 3 + 2 + 2 + 1: [2,1,1], [2,2], [3,1], [4].
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a:= proc(k) option remember; local b; b:=
proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
add((l-> l+[0, l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, k))))
end: b(2*k$2)[2]
end:
seq(a(n), n=0..37);