A209817
Number of partitions of 3n in which every part is
Original entry on oeis.org
0, 1, 5, 19, 54, 141, 331, 733, 1527, 3060, 5888, 11004, 19978, 35452, 61538, 104875, 175618, 289656, 470914, 755880, 1198693, 1880246, 2918919, 4488553, 6840398, 10337947, 15500575, 23070000, 34094908, 50055877, 73026093, 105902689, 152706404, 219004225
Offset: 1
The 5 partitions of 9 with parts <3 are as follows:
2+2+2+2+1
2+2+2+1+1+1
2+2+1+1+1+1+1
2+1+1+1+1+1+1+1
1+1+1+1+1+1+1+1+1.
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, b(n-i, i))))
end:
a:= n-> b(3*n, n-1):
seq(a(n), n=1..50); # Alois P. Heinz, Jul 09 2012
-
f[n_] := Length[Select[IntegerPartitions[3 n], First[#] <= n - 1 &]]; Table[f[n], {n, 1, 25}] (* A209817 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[3*n, n-1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)
A304134
Number of partitions of 5n into exactly n parts.
Original entry on oeis.org
1, 1, 5, 19, 64, 192, 532, 1367, 3319, 7657, 16928, 36043, 74287, 148702, 290071, 552767, 1031391, 1887776, 3395084, 6007963, 10474462, 18010859, 30574655, 51284587, 85064661, 139620591, 226914505, 365371100, 583164222, 923075291, 1449643115, 2259616844
Offset: 0
n | Partitions of 5n into exactly n parts
--+------------------------------------------------
1 | 5;
2 | 9+1, 8+2, 7+3, 6+4, 5+5;
3 | 13+1+1, 12+2+1, 11+3+1, 11+2+2, 10+4+1, 10+3+2,
| 9+5+1, 9+4+2, 9+3+3, 8+6+1, 8+5+2, 8+4+3,
| 7+7+1, 7+6+2, 7+5+3, 7+4+4, 6+6+3, 6+5+4,
| 5+5+5;
====================================================================
n | Partitions of 4n in which every part is <=n.
--+-----------------------------------------------------------------
1 | 1+1+1+1;
2 | 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1;
3 | 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1,
| 3+3+2+1+1+1+1, 3+3+1+1+1+1+1+1, 3+2+2+2+2+1, 3+2+2+2+1+1+1,
| 3+2+2+1+1+1+1+1, 3+2+1+1+1+1+1+1+1, 3+1+1+1+1+1+1+1+1+1,
| 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+1+1+1+1+1+1,
| 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,
| 1+1+1+1+1+1+1+1+1+1+1+1;
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) +b(n-i, min(i, n-i)))
end:
a:= n-> b(4*n, n):
seq(a(n), n=0..35); # Alois P. Heinz, May 07 2018
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b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]];
a[n_] := b[4n, n];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)
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{a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(4*n)))), 4*n)}
Showing 1-2 of 2 results.
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