A222044
Sum of smallest parts of all partitions of n into an odd number of parts.
Original entry on oeis.org
0, 1, 2, 4, 5, 8, 11, 15, 19, 28, 35, 47, 61, 80, 102, 136, 168, 218, 276, 350, 437, 556, 686, 860, 1063, 1321, 1620, 2005, 2443, 2998, 3649, 4445, 5377, 6531, 7863, 9496, 11398, 13694, 16373, 19603, 23347, 27834, 33058, 39259, 46467, 55020, 64914, 76599
Offset: 0
a(6) = 11: partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6], sum of smallest parts is 1+2+1+1+6 = 11.
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b:= proc(n, i) option remember;
[`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
`if`(n [l[2], l[1]])(b(n-i, i))))
end:
a:= n-> b(n, n)[1]:
seq(a(n), n=0..60);
-
b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 03 2017, translated from Maple *)
Table[Total[Min/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* Harvey P. Dale, Jul 05 2019 *)
A222045
Sum of smallest parts of all partitions of n into an even number of parts.
Original entry on oeis.org
0, 0, 1, 1, 4, 4, 9, 10, 19, 21, 34, 40, 62, 72, 103, 124, 173, 207, 279, 337, 445, 538, 694, 842, 1077, 1299, 1634, 1977, 2464, 2969, 3669, 4411, 5410, 6488, 7896, 9447, 11442, 13640, 16421, 19536, 23411, 27761, 33124, 39174, 46554, 54915, 65008, 76485, 90258
Offset: 0
a(6) = 9: partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1], sum of smallest parts is 1+1+1+3+2+1 = 9.
-
b:= proc(n, i) option remember;
[`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
`if`(n [l[2], l[1]])(b(n-i, i))))
end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);
-
b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[nJean-François Alcover, Feb 03 2017, translated from Maple *)
A222047
Sum of largest parts of all partitions of n into an odd number of parts.
Original entry on oeis.org
0, 1, 2, 4, 6, 11, 17, 28, 41, 66, 93, 140, 195, 282, 384, 541, 722, 992, 1311, 1762, 2299, 3045, 3929, 5127, 6559, 8458, 10726, 13689, 17225, 21780, 27224, 34134, 42387, 52769, 65138, 80544, 98887, 121538, 148456, 181456, 220590, 268252, 324677, 392961
Offset: 0
a(6) = 17: partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6], sum of largest parts is 2+2+3+4+6 = 17.
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b:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
`if`(i>n, [0, 0], b(n, i+1)+(l-> [l[2], l[1]])(b(n-i, i)))
end:
a:= n-> b(n,1)[1]:
seq(a(n), n=0..50);
-
Table[Total[Max[#]&/@Select[IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,50}] (* Harvey P. Dale, Apr 19 2014 *)
b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, b[n, i+1] + Reverse[b[n-i, i]]]; a[n_] := b[n, 1][[1]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)
A222048
Sum of largest parts of all partitions of n into an even number of parts.
Original entry on oeis.org
0, 0, 1, 2, 6, 9, 18, 26, 45, 62, 99, 135, 204, 274, 396, 527, 741, 973, 1333, 1736, 2331, 3007, 3970, 5079, 6615, 8393, 10796, 13605, 17320, 21673, 27339, 34001, 42540, 52597, 65324, 80332, 99127, 121274, 148745, 181131, 220956, 267852, 325114, 392476, 474178
Offset: 0
a(6) = 18: partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1], sum of largest parts is 1+2+3+3+4+5 = 18.
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b:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
`if`(i>n, [0, 0], b(n, i+1)+(l-> [l[2], l[1]])(b(n-i, i)))
end:
a:= n-> b(n,1)[2]:
seq(a(n), n=0..50);
-
b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, b[n, i+1] + Reverse @ b[n-i, i]]; a[n_] := b[n, 1][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
A222049
Difference between sums of largest parts of all partitions of n into odd number of parts and into even number of parts.
Original entry on oeis.org
0, 1, 1, 2, 0, 2, -1, 2, -4, 4, -6, 5, -9, 8, -12, 14, -19, 19, -22, 26, -32, 38, -41, 48, -56, 65, -70, 84, -95, 107, -115, 133, -153, 172, -186, 212, -240, 264, -289, 325, -366, 400, -437, 485, -544, 597, -649, 714, -799, 869, -942, 1037, -1148, 1246, -1351
Offset: 0
a(6) = -1 = (2+2+3+4+6) - (1+2+3+3+4+5) because the partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6] and the partitions of 6 into an even number of parts are [1,1,1,1,1,1], [2,2,1,1], [3,1,1,1], [3,3], [4,2], [5,1].
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b:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
`if`(i>n, [0, 0], b(n, i+1)+(l-> [l[2], l[1]])(b(n-i, i)))
end:
a:= n-> (l->l[1]-l[2])(b(n, 1)):
seq(a(n), n=0..60);
-
b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, b[n, i+1] + Reverse @ b[n-i, i]]; a[n_] := b[n, 1][[1]]-b[n, 1][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
Showing 1-5 of 5 results.
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