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 *)
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 *)
A237799
Number of partitions of n such that 2*(least part) > number of parts.
Original entry on oeis.org
1, 1, 1, 2, 2, 4, 4, 6, 7, 9, 10, 14, 15, 19, 23, 28, 32, 40, 46, 56, 65, 77, 89, 107, 122, 143, 165, 193, 220, 257, 292, 338, 385, 443, 503, 578, 653, 746, 844, 962, 1083, 1231, 1384, 1567, 1761, 1987, 2227, 2510, 2807, 3153, 3523, 3949, 4403, 4927, 5485
Offset: 1
a(7) = 4 counts these partitions: 7, 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 *)
A118082
Number of partitions of n such that largest part k occurs floor(k/2) times.
Original entry on oeis.org
1, 0, 1, 2, 2, 3, 3, 4, 5, 6, 8, 10, 12, 15, 19, 22, 27, 32, 39, 45, 54, 63, 75, 87, 102, 118, 139, 160, 186, 214, 248, 284, 328, 375, 430, 490, 561, 637, 727, 824, 935, 1058, 1199, 1352, 1528, 1720, 1938, 2177, 2448, 2743, 3079, 3445, 3856, 4307, 4813, 5365, 5985
Offset: 0
a(8)=5 because we have [4,4],[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*floor(k/2))/product(1-x^j,j=1..k-1),k=1..15): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60);
A118083
Number of partitions of n such that largest part k occurs at least floor(k/2) times.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 26, 30, 38, 45, 55, 64, 79, 91, 110, 128, 152, 176, 209, 240, 282, 325, 379, 434, 505, 576, 666, 760, 873, 993, 1139, 1290, 1473, 1668, 1897, 2141, 2430, 2736, 3095, 3481, 3925, 4404, 4958, 5550, 6232, 6968, 7805, 8710
Offset: 0
a(8)=11 because we have [4,4],[3,3,2],[3,3,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1],[2,2,2,2],[2,2,2,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1].
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g:=sum(x^(k*floor(k/2))/product(1-x^j,j=1..k),k=1..15): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60);
Showing 1-5 of 5 results.
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