A222048 -id:A222048 - cp's OEIS frontend

A211373 Sum of median parts of all partitions of n into an odd number of parts.

0, 1, 2, 4, 5, 9, 12, 18, 24, 37, 47, 68, 89, 123, 160, 218, 276, 370, 472, 615, 778, 1006, 1259, 1607, 2005, 2530, 3136, 3926, 4833, 6004, 7363, 9070, 11067, 13562, 16461, 20053, 24241, 29370, 35362, 42648, 51135, 61400, 73367, 87718, 104448, 124428, 147655

Offset: 0

Alois P. Heinz, Feb 06 2013

nonn

			a(6) = 12: 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 median parts is 1+2+2+1+6 = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A222044, A222045, A222047, A222048, A268359.

with(combinat):
a:= n-> add(`if`(nops(l)::odd, l[(nops(l)+1)/2], 0), l=partition(n)):
seq(a(n), n=0..40);

a[n_] := Sum[If[OddQ @ Length[l], l[[(Length[l]+1)/2]], 0], {l, IntegerPartitions[n]}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)

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

Alois P. Heinz, Feb 06 2013

nonn

a(n) + A222045(n) = A046746(n).

a(n) - A222045(n) = A222046(n).

			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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A046746, A211373, A222045, A222046, A222047, A222048, A222049.

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 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 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

Alois P. Heinz, Feb 06 2013

nonn

A222044(n) + a(n) = A046746(n).

A222044(n) - a(n) = A222046(n).

			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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A046746, A211373, A222044, A222046, A222047, A222048, A222049.

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 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 06 2019

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

Alois P. Heinz, Feb 06 2013

nonn

a(n) + A222048(n) = A006128(n).

a(n) - A222048(n) = A222049(n).

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A211373, A222044, A222045, A222046, A222048, A222049.

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 *)

A222046 Difference between sums of smallest 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, 3, 1, 4, 2, 5, 0, 7, 1, 7, -1, 8, -1, 12, -5, 11, -3, 13, -8, 18, -8, 18, -14, 22, -14, 28, -21, 29, -20, 34, -33, 43, -33, 49, -44, 54, -48, 67, -64, 73, -66, 85, -87, 105, -94, 114, -120, 132, -128, 156, -159, 174, -172, 203, -213, 234, -232, 263

Offset: 0

Alois P. Heinz, Feb 06 2013

sign

			a(6) = 2 = (1+2+1+1+6) - (1+1+1+3+2+1) 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].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A211373, A222044, A222045, A222047, A222048, A222049.

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-> (l->l[1]-l[2])(b(n, n)):
seq(a(n), n=0..100);

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, Jan 23 2017, translated from Maple *)

a(n) = A222044(n) - A222045(n).

a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 06 2019

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

Alois P. Heinz, Feb 06 2013

sign

			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].

Alois P. Heinz, Table of n, a(n) for n = 0..6000

Cf. A211373, A222044, A222045, A222046, A222047, A222048.

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 *)

a(n) = A222047(n) - A222048(n).

G.f.: Sum_{i>=0} i*x^i/Product_{j=1..i} (1 + x^j). - Ilya Gutkovskiy, Apr 13 2018

A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 9, 16, 26, 41, 65, 95, 142, 202, 293, 403, 568, 766, 1054, 1399, 1886, 2469, 3276, 4237, 5538, 7094, 9162, 11628, 14856, 18704, 23670, 29590, 37130, 46109, 57428, 70885, 87685, 107634, 132324, 161595, 197545, 240091, 291990, 353302, 427624

Offset: 0

Alois P. Heinz, Feb 13 2013

nonn

			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], difference between sum of largest parts and sum of smallest parts is (1+2+3+3+4+5) - (1+1+1+3+2+1) = 18 - 9 = 9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A116686, A211870, A222045, A222048.

g:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
      `if`(i>n, [0, 0], g(n, i+1)+(l-> [l[2], l[1]])(g(n-i, i)))
    end:
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-> g(n, 1)[2] -b(n, n)[2]:
seq(a(n), n=0..50);

g[n_, i_] := g[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, g[n, i+1] + Reverse[g[n-i, i]]]; 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 15 2017, translated from Maple *)

a(n) = A222048(n) - A222045(n).

a(n) = A116686(n) - A211870(n).

cp's OEIS Frontend

A211373 Sum of median parts of all partitions of n into an odd number of parts.

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A222044 Sum of smallest parts of all partitions of n into an odd number of parts.

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A222045 Sum of smallest parts of all partitions of n into an even number of parts.

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A222047 Sum of largest parts of all partitions of n into an odd number of parts.

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A222046 Difference between sums of smallest parts of all partitions of n into odd number of parts and into even number of parts.

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A222049 Difference between sums of largest parts of all partitions of n into odd number of parts and into even number of parts.

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A211881 Difference between sum of largest parts and sum of smallest parts of all partitions of n into an even number of parts.

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