A222730 -id:A222730 - cp's OEIS frontend

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

0, 1, 2, 6, 10, 21, 33, 59, 89, 145, 212, 325, 463, 680, 948, 1348, 1845, 2558, 3446, 4681, 6219, 8306, 10901, 14352, 18632, 24230, 31151, 40077, 51074, 65088, 82290, 103986, 130517, 163679, 204078, 254174, 314975, 389839, 480369, 591133, 724600, 886965

Offset: 0

Vladeta Jovovic, Mar 25 2005

Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).

Equals A000041 convolved with A026741. - Gary W. Adamson, Jun 11 2009

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.

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

Cf. A026741. - Gary W. Adamson, Jun 11 2009

Column k=1 of A222730. - Alois P. Heinz, Mar 03 2013

gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
       l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}]  (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).
a(n) = A066186(n) - A194544(n). - Omar E. Pol, Nov 20 2011
a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016

More terms from James Sellers, Apr 22 2005

A117525 Total sum of parts of multiplicity 2 in all partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 7, 9, 20, 22, 44, 56, 90, 119, 186, 236, 355, 461, 651, 848, 1177, 1506, 2050, 2626, 3482, 4443, 5823, 7353, 9524, 11983, 15307, 19163, 24277, 30174, 37920, 46925, 58463, 72006, 89155, 109209, 134418, 163973, 200605, 243700, 296696, 358862

Offset: 0

Vladeta Jovovic, Apr 26 2006

For m > 0, column m of A222730 is asymptotic to sqrt(3) * (2*m+1) * exp(Pi*sqrt(2*n/3)) / (2 * m^2 * (m+1)^2 * Pi^2) ~ 6 * (2*m+1) * n * p(n) / (m^2 * (m+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018

			a(5) = 3 because the partitions of 5 that have parts with multiplicity 2 are [3,1,1] and [2,2,1] and the sum of those parts is 1+2 = 3.

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

Cf. A103628.

Column k=2 of A222730. - Alois P. Heinz, Mar 03 2013

g:=(x^2/(1-x^2)^2-x^3/(1-x^3)^2)/Product(1-x^i,i=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, May 13 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [l[1], l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[4]]; a[0] = a[1] = 0; Table[a[n], {n, 0, 50}]   (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f. for total sum of parts of multiplicity m in all partitions of n is (x^m/(1-x^m)^2-x^(m+1)/(1-x^(m+1))^2)/Product(1-x^i,i=1..infinity).

a(n) ~ 5 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (72 * Pi^2). - Vaclav Kotesovec, May 29 2018

More terms from Emeric Deutsch, May 13 2006

A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.

Original entry on oeis.org

0, 0, 3, 11, 36, 79, 186, 345, 672, 1163, 2026, 3273, 5388, 8301, 12912, 19349, 28961, 42071, 61253, 86921, 123404, 171972, 239020, 327386, 447743, 604255, 813645, 1084657, 1441643, 1899450, 2496510, 3255653, 4234822, 5472953, 7053217, 9038784, 11554020

Offset: 0

Alois P. Heinz, Mar 04 2013

nonn

			The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4].  Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, thus a(4) = 36.

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

Column k=0 of A222730.

Cf. A000041, A000217, A014153.

b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0], `if`(p<1, [0$2],
      add((l->`if`(m=0, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..55);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

a(n) = A000217(n)*A000041(n)-A014153(n-1).

A222731 Total sum of parts of multiplicity 3 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 3, 4, 4, 11, 13, 21, 30, 44, 59, 92, 115, 165, 225, 305, 394, 546, 700, 931, 1204, 1572, 2005, 2613, 3290, 4218, 5328, 6745, 8423, 10630, 13193, 16475, 20386, 25269, 31072, 38346, 46882, 57478, 70066, 85415, 103582, 125794, 151916, 183576, 220962

Offset: 3

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=3, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=3..50);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[5]]; Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^3/(1-x^3)^2-x^4/(1-x^4)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 7 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (288 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222732 Total sum of parts of multiplicity 4 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 4, 4, 6, 8, 16, 19, 30, 36, 59, 73, 106, 135, 191, 242, 331, 420, 569, 712, 941, 1183, 1546, 1931, 2476, 3087, 3933, 4872, 6137, 7568, 9471, 11629, 14427, 17647, 21758, 26499, 32470, 39393, 48030, 58028, 70385, 84749, 102348, 122794, 147633, 176554

Offset: 4

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=4, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=4..55);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[6]]; Table[a[n], {n, 4, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^4/(1-x^4)^2-x^5/(1-x^5)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 9 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (800 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222733 Total sum of parts of multiplicity 5 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 6, 6, 11, 14, 23, 29, 43, 52, 76, 100, 135, 174, 235, 294, 397, 500, 651, 821, 1060, 1324, 1692, 2107, 2658, 3297, 4139, 5089, 6339, 7778, 9604, 11746, 14425, 17533, 21427, 25960, 31548, 38080, 46070, 55375, 66718, 79957, 95906, 114555

Offset: 5

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=5, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=5..55);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[7]]; Table[a[n], {n, 5, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^5/(1-x^5)^2-x^6/(1-x^6)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 11 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (1800 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222734 Total sum of parts of multiplicity 6 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 9, 12, 18, 22, 36, 43, 62, 77, 107, 133, 186, 229, 306, 384, 499, 621, 810, 999, 1277, 1582, 1997, 2453, 3088, 3776, 4698, 5742, 7088, 8618, 10592, 12824, 15654, 18910, 22955, 27615, 33400, 40028, 48174, 57593, 69018, 82231, 98225

Offset: 6

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=6, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=6..55);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[8]]; Table[a[n], {n, 6, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^6/(1-x^6)^2-x^7/(1-x^7)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 13 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (3528 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222735 Total sum of parts of multiplicity 7 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 6, 9, 10, 16, 20, 29, 36, 53, 66, 91, 112, 152, 190, 251, 315, 409, 510, 655, 809, 1029, 1271, 1602, 1967, 2457, 3009, 3729, 4543, 5595, 6801, 8321, 10069, 12258, 14783, 17906, 21511, 25947, 31073, 37315, 44542, 53285, 63415, 75587, 89687

Offset: 7

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=7, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=7..60);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[9]]; Table[a[n], {n, 7, 60}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^7/(1-x^7)^2-x^8/(1-x^8)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 15 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (6272 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222736 Total sum of parts of multiplicity 8 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 9, 10, 14, 18, 27, 32, 46, 57, 80, 99, 134, 163, 219, 270, 350, 433, 561, 686, 875, 1074, 1349, 1652, 2062, 2509, 3116, 3783, 4650, 5633, 6893, 8305, 10108, 12153, 14709, 17630, 21243, 25371, 30452, 36271, 43335, 51478, 61311, 72598

Offset: 8

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=8, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=8..60);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[10]]; Table[a[n], {n, 8, 60}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^8/(1-x^8)^2-x^9/(1-x^9)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 17 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (10368 * Pi^2). - Vaclav Kotesovec, May 29 2018

A222737 Total sum of parts of multiplicity 9 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 10, 14, 16, 25, 30, 42, 53, 71, 88, 121, 148, 195, 241, 312, 384, 494, 605, 765, 943, 1179, 1441, 1796, 2181, 2694, 3273, 4011, 4849, 5922, 7130, 8652, 10398, 12552, 15021, 18072, 21558, 25816, 30729, 36649, 43480, 51705, 61163

Offset: 9

Alois P. Heinz, Mar 03 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A222730.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=9, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=9..60);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[11]]; Table[a[n], {n, 9, 60}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: (x^9/(1-x^9)^2-x^10/(1-x^10)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 19 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (16200 * Pi^2). - Vaclav Kotesovec, May 29 2018

cp's OEIS Frontend

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

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A117525 Total sum of parts of multiplicity 2 in all partitions of n.

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A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.

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A222731 Total sum of parts of multiplicity 3 in all partitions of n.

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A222732 Total sum of parts of multiplicity 4 in all partitions of n.

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A222733 Total sum of parts of multiplicity 5 in all partitions of n.

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A222734 Total sum of parts of multiplicity 6 in all partitions of n.

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A222735 Total sum of parts of multiplicity 7 in all partitions of n.

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