A103628 -id:A103628 - cp's OEIS frontend

A222730 Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

0, 0, 1, 3, 2, 1, 11, 6, 0, 1, 36, 10, 3, 0, 1, 79, 21, 3, 1, 0, 1, 186, 33, 7, 3, 1, 0, 1, 345, 59, 9, 4, 1, 1, 0, 1, 672, 89, 20, 4, 4, 1, 1, 0, 1, 1163, 145, 22, 11, 4, 2, 1, 1, 0, 1, 2026, 212, 44, 13, 6, 4, 2, 1, 1, 0, 1, 3273, 325, 56, 21, 8, 6, 2, 2, 1, 1, 0, 1

Offset: 0

Alois P. Heinz, Mar 03 2013

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

			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, for m=1 the sum is 2+(3+1)+4 = 10, for m=2 the sum is 1+2 = 3, for m=3 the sum is 0, for m=4 the sum is 1 => row 4 = [36, 10, 3, 0, 1].
Triangle T(n,k) begins:
    0;
    0,  1;
    3,  2,  1;
   11,  6,  0, 1;
   36, 10,  3, 0, 1;
   79, 21,  3, 1, 0, 1;
  186, 33,  7, 3, 1, 0, 1;
  345, 59,  9, 4, 1, 1, 0, 1;
  672, 89, 20, 4, 4, 1, 1, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738.

Cf. A000041, A000217, A002865, A014153, A066186.

b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0],
      `if`(p=0, [0$(n+2)], add((l-> subsop(m+2=p*l[1]+l[m+2], l))
          ([b(n-p*m, p-1)[], 0$(p*m)]), m=0..n/p)))
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=0..14);

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}]]]; Rest /@ Table[b[n, n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

Sum_{k=0..n} k*T(n,k) = A066186(n) = n*A000041(n).
Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0.
Sum_{k=0..n} T(n,k) = n*(n+1)/2*A000041(n) = A000217(n)*A000041(n).
(2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0.
T(2*n+1,n+1) = A002865(n).

A194544 Total sum of repeated parts in all partitions of n.

Original entry on oeis.org

0, 0, 2, 3, 10, 14, 33, 46, 87, 125, 208, 291, 461, 633, 942, 1292, 1851, 2491, 3484, 4629, 6321, 8326, 11143, 14513, 19168, 24720, 32185, 41193, 53030, 67297, 85830, 108116, 136651, 171040, 214462, 266731, 332197, 410730, 508201, 625082, 768920, 940938

Offset: 0

Omar E. Pol, Nov 19 2011

nonn

			For n = 6 we have:
--------------------------------------
.                          Sum of
Partitions             repeated parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 0
2 + 2 + 2 .................. 6
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 6
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
--------------------------------------
Total ..................... 33
So a(6) = 33.

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

Cf. A066186, A103628, A135010, A138121, A194452.

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*i*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Nov 20 2011

b[n_, i_] := b[n, i] = Module[{h, j, t}, Which [n<0, {0, 0}, n==0, {1, 0}, i<1, {0, 0}, True, h = {0, 0}; For[j=0, j <= Quotient[n, i], j++, t = b[n - i*j, i-1]; h = {h[[1]] + t[[1]], h[[2]] + t[[2]] + If[j<2, 0, t[[1]]* i*j]}]; h]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)
Table[Total[Flatten[Select[Flatten[Split/@IntegerPartitions[n],1], Length[ #]> 1&]]],{n,0,50}] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = A066186(n) - A103628(n), n >= 1.

a(n) ~ exp(sqrt(2*n/3)*Pi) * (1/(4*sqrt(3))-3*sqrt(3)/(8*Pi^2)) * (1 - Pi*(135+2*Pi^2)/(24*(2*Pi^2-9)*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016

More terms from Alois P. Heinz, Nov 20 2011

A276423 Sum of the odd singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

Original entry on oeis.org

0, 1, 0, 4, 4, 13, 13, 33, 41, 79, 98, 171, 223, 354, 458, 692, 905, 1306, 1694, 2375, 3077, 4202, 5401, 7238, 9260, 12200, 15495, 20145, 25446, 32686, 41020, 52170, 65117, 82071, 101852, 127374, 157277, 195289, 239915, 296023, 362000, 444063, 540595, 659662

Offset: 0

Emeric Deutsch, Sep 14 2016

nonn

			a(4) = 4 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the odd singletons are 0,0,0,4,0, respectively; their sum is 4.
a(5) = 13 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the odd singletons are 0,0,1,3,3,1,5, respectively; their sum is 13.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A103628, A265257, A276422, A276424, A276425.

g := x*(1-x+3*x^2+3*x^4-x^5+x^6)/((1-x^4)^2*(product(1-x^i, i = 1..120))): gser := series(g, x = 0, 60); seq(coeff(gser, x, n), n = 0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, add((p-> p+`if`(i::odd and j=1,
      [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 14 2016

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, Sum[Function[p, p + If[OddQ[i] && j == 1, {0, If[p === 0, 0, i*p[[1]]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 04 2016 after Alois P. Heinz *)
Table[Total[Select[Flatten[Tally/@IntegerPartitions[n],1],#[[2]]==1 && OddQ[ #[[1]]]&][[All,1]]],{n,0,50}] (* Harvey P. Dale, May 25 2018 *)

G.f.: g(x) = x*(1-x+3*x^2+3*x^4-x^5+x^6)/((1-x^4)^2*Product_{j>=1} 1-x^j).
a(n) = Sum_{k>=0} k*A276422(n,k).
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (16*Pi^2). - Vaclav Kotesovec, Jun 12 2025

A276425 Sum of the even singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

Original entry on oeis.org

0, 0, 2, 2, 6, 8, 20, 26, 48, 66, 114, 154, 240, 326, 490, 656, 940, 1252, 1752, 2306, 3142, 4104, 5500, 7114, 9372, 12030, 15656, 19932, 25628, 32402, 41270, 51816, 65400, 81608, 102226, 126800, 157698, 194550, 240454, 295110, 362600, 442902, 541342, 658230

Offset: 0

Emeric Deutsch, Sep 14 2016

nonn

			a(4) = 6 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively; their sum is 6.
a(5) = 8 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0 respectively; their sum is 8.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A103628, A276422, A276423, A276424.

g := 2*x^2*(1+x^2+x^4)/((1-x^4)^2*(product(1-x^i, i = 1 .. 120))): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, add((p-> p+`if`(i::even and j=1,
      [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 14 2016

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, 0, Sum[Function[p, p + If[EvenQ[i] && j == 1, {0, i*p[[1]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)

G.f.:  g(x) = 2x^2*(1+x^2+x^4)/((1-x^4)^2 product(1-x^j, j>=1)).
a(n) = Sum(k*A276424(n,k), k>=0).
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (16*Pi^2). - Vaclav Kotesovec, Jun 12 2025

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

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Original entry on oeis.org

0, 1, 3, 7, 16, 28, 59, 91, 170, 269, 450, 655, 1162, 1602, 2527, 3793, 5805, 8034, 12660, 17131, 26484, 37384, 53738, 73504, 114683, 153613, 221225, 313339, 453769, 609179, 927968, 1223909, 1804710, 2522264, 3539835, 4855420, 7439870, 9765555, 14009545

Offset: 0

Alois P. Heinz, Feb 27 2013

nonn

			a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

From Vaclav Kotesovec, May 24 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0
c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1
c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2
(End)

A264403 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 0, 3, 4, 0, 1, 1, 1, 0, 4, 2, 2, 1, 2, 1, 2, 0, 5, 6, 0, 2, 1, 3, 2, 2, 0, 6, 5, 2, 1, 4, 1, 4, 2, 3, 0, 8, 9, 1, 3, 2, 5, 2, 4, 3, 3, 0, 10, 7, 3, 3, 6, 2, 7, 2, 6, 3, 5, 0, 12, 16, 0, 4, 4, 7, 3, 8, 3, 7, 5, 5, 0, 15, 11, 6, 4, 8, 5, 9, 3, 12, 3, 10, 5, 7, 0, 18

Offset: 0

Emeric Deutsch, Nov 27 2015

Row n contains n+1 entries (n>=0).
Row sums yield the partition numbers (A000041).
T(n,0) = A007690(n).
T(n,n) = A000009(n).
Sum_{k>=0} k*T(n,k) = A103628(n).

			T(7,5) = 2 because we have [3,2,1,1] and [5,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,2;
2,0,1,0,2;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000009, A000041, A007690, A103628.

g := product(1+t^j*x^j+x^(2*j)/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 25)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(expand(b(n-i*j, i-1)*
      `if`(j=1, x^i, 1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Nov 27 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[b[n-i*j, i-1]*If[j == 1, x^i, 1]], {j, 0, n/i}]]]; T[n_] := Function[p,Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

G.f.: G(t,x) = Product_{j>=1} (1+t^j*x^j + x^{2*j}/(1 - x^j)).

cp's OEIS Frontend

A222730 Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A194544 Total sum of repeated parts in all partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A276423 Sum of the odd singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A276425 Sum of the even singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264403 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs