A237800
Number of partitions of n such that 2*(least part) >= number of parts.
Original entry on oeis.org
1, 2, 2, 3, 3, 5, 5, 8, 9, 12, 14, 19, 21, 27, 32, 39, 45, 56, 64, 78, 90, 107, 124, 148, 169, 199, 229, 268, 306, 357, 406, 471, 536, 617, 701, 805, 910, 1041, 1177, 1341, 1511, 1717, 1931, 2187, 2457, 2773, 3109, 3503, 3918, 4403, 4919, 5514, 6150, 6881
Offset: 1
a(7) = 5 counts these partitions: 7, 61, 52, 43, 322.
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z = 55; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] < t[p]], {n, z}] (* A237758 *)
Table[Count[q[n], p_ /; 2 Min[p] == t[p]], {n, z}] (* A237757 *)
Table[Count[q[n], p_ /; 2 Min[p] > t[p]], {n, z}] (* A237799 *)
Table[Count[q[n], p_ /; 2 Min[p] >= t[p]], {n, z}] (* A237800 *)
A118084
Number of partitions of n such that largest part k occurs at most floor(k/2) times.
Original entry on oeis.org
0, 1, 2, 3, 5, 7, 11, 16, 23, 33, 46, 63, 86, 116, 153, 203, 265, 345, 444, 571, 727, 925, 1166, 1468, 1836, 2293, 2845, 3525, 4345, 5347, 6550, 8011, 9758, 11867, 14380, 17399, 20984, 25269, 30341, 36376, 43500, 51943, 61877, 73608, 87373, 103571
Offset: 1
a(8)=16 because we have [8],[7,1],[6,2],[6,1,1],[5,3],[5,2,1],[5,1,1,1],[4,4],[4,3,1],[4,2,2],[4,2,1,1],[4,1,1,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1] and [2,1,1,1,1,1,1].
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g:=sum(x^k*(1-x^(k*(floor(k/2))))/product(1-x^j,j=1..k),k=1..85): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50);
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z=55 ; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n,z}] (* Clark Kimberling, Feb 15 2014 *)
A237758
Number of partitions of n such that 2*(least part) < number of parts.
Original entry on oeis.org
0, 0, 1, 2, 4, 6, 10, 14, 21, 30, 42, 58, 80, 108, 144, 192, 252, 329, 426, 549, 702, 895, 1131, 1427, 1789, 2237, 2781, 3450, 4259, 5247, 6436, 7878, 9607, 11693, 14182, 17172, 20727, 24974, 30008, 35997, 43072, 51457, 61330, 72988, 86677, 102785, 121645
Offset: 1
a(5) = 4 counts these partitions: 311, 221, 2111, 11111.
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z = 55; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] < t[p]], {n, z}] (* A237758 *)
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n, z}] (* A118084 *)
Table[Count[q[n], p_ /; 2 Min[p] == t[p]], {n, z}] (* A237757 *)
Table[Count[q[n], p_ /; 2 Min[p] > t[p]], {n, z}] (* A237799 *)
Table[Count[q[n], p_ /; 2 Min[p] >= t[p]], {n, z}] (* A237800 *)
Showing 1-3 of 3 results.
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