A197126 -id:A197126 - cp's OEIS frontend

A116646 Number of doubletons in all partitions of n. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

0, 0, 1, 0, 2, 2, 4, 5, 10, 11, 20, 25, 38, 49, 73, 91, 131, 167, 228, 291, 392, 493, 653, 822, 1065, 1336, 1714, 2131, 2706, 3354, 4209, 5193, 6471, 7934, 9817, 11990, 14725, 17909, 21875, 26477, 32172, 38797, 46893, 56339, 67804, 81147, 97260, 116017

Offset: 0

Emeric Deutsch, Feb 20 2006

nonn

a(n) = (the number of 2's in all partitions of n) - (the number of 3's in all partitions of n). - Gregory L. Simay, Jul 28 2020

			a(6) = 4 because in the partitions of 6, namely [6],[5,1],[4,2],[4,(1,1)],[(3,3)],[3,2,1],[3,1,1,1],[2,2,2],[(2,2),(1,1)],[2,1,1,1,1] and [1,1,1,1,1,1], we have a total of 4 doubletons (shown between parentheses).

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

Cf. A116644. Column k=2 of A197126.

f:= x^2/(1+x)/(1-x^3)/product(1-x^j, j=1..70): fser:= series(f,x=0,70): seq(coeff(fser,x,n), n=0..55);

nmax = 50; CoefficientList[Series[x^2/((1+x)*(1-x^3)) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Sum[PartitionsP[n-6*m-2] - PartitionsP[n-6*m-3] + PartitionsP[n-6*m-4], {m, 0, Floor[n/6]}], {n, 0, 50}] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: x^2 / ((1+x)*(1-x^3)*(Product_{j>=1} 1-x^j)).
a(n) = Sum_{k>=0} k * A116644(n,k).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Mar 07 2016

A117524 Total number of parts of multiplicity 3 in all partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 3, 7, 8, 13, 17, 25, 32, 48, 59, 83, 108, 145, 183, 247, 310, 406, 512, 659, 824, 1055, 1307, 1651, 2047, 2558, 3146, 3913, 4788, 5904, 7202, 8821, 10707, 13054, 15770, 19118, 23027, 27775, 33312, 40029, 47835, 57231, 68182, 81261

Offset: 1

Vladeta Jovovic, Apr 26 2006

			a(9) = 7 because among the 30 (=A000041(9)) partitions of 9 only [6,(1,1,1)],[4,2,(1,1,1)],[(3,3,3)],[3,3,(1,1,1)],[3,(2,2,2)],[(2,2,2),(1,1,1)] contain parts of multiplicity 3 and their total number is 7 (shown between parentheses)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A024786, A116646. Column k=3 of A197126.

g:=(x^3/(1-x^3)-x^4/(1-x^4))/product(1-x^i,i=1..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),n=1..58); # Emeric Deutsch, Apr 29 2006

G.f. for total number of parts of multiplicity m in all partitions of n is (x^m/(1-x^m)-x^(m+1)/(1-x^(m+1)))/Product(1-x^i,i=1..infinity).
a(n) = Sum(k*A118806(n,k), k>=0). - Emeric Deutsch, Apr 29 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (24*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 9, 3, 1, 1, 17, 5, 2, 1, 1, 28, 9, 4, 2, 1, 1, 47, 14, 7, 3, 2, 1, 1, 73, 24, 10, 6, 3, 2, 1, 1, 114, 35, 17, 9, 5, 3, 2, 1, 1, 170, 55, 25, 14, 8, 5, 3, 2, 1, 1, 253, 80, 38, 20, 13, 7, 5, 3, 2, 1, 1, 365, 118, 55, 31, 18, 12, 7, 5, 3, 2, 1, 1, 525, 167, 80, 44, 27, 17, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Apr 04 2018

Column k gives the partial sums of the k-th column of triangle A197126.

			Triangle begins:
    1;
    2,   1;
    5,   1,  1;
    9,   3,  1,  1;
   17,   5,  2,  1,  1;
   28,   9,  4,  2,  1,  1;
   47,  14,  7,  3,  2,  1,  1;
   73,  24, 10,  6,  3,  2,  1,  1;
  114,  35, 17,  9,  5,  3,  2,  1,  1;
  170,  55, 25, 14,  8,  5,  3,  2,  1,  1;
  253,  80, 38, 20, 13,  7,  5,  3,  2,  1,  1;
  365, 118, 55, 31, 18, 12,  7,  5,  3,  2,  1,  1;
  525, 167, 80, 44, 27, 17, 11,  7,  5,  3,  2,  1,  1;
...

Alois P. Heinz, Rows n = 1..150, flattened

Column 1 gives A000097.
Row sums give A014153.
Cf. A000041, A000070, A197126, A284870.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
    end:
T:= proc(n, k) option remember;
      b(n$2, k)[2]+`if`(nAlois P. Heinz, Apr 27 2018

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m==k, l+{0, l[[1]]}, l]][b[n-p*m, p-1, k]], {m, 0, n/p}]]];
T[n_, k_] := b[n, n, k][[2]] + If[n < k, 0, T[n-1, k]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

T(n,k) = Sum_{j=k..n} A197126(j,k).
T(2n+1,n+1) = A000041(n). - Alois P. Heinz, Apr 27 2018
Sum_{k=1..n} k * T(n,k) = A284870(n). - Alois P. Heinz, May 14 2018

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)

A222704 Total number of parts of multiplicity 4 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 11, 13, 20, 24, 37, 45, 64, 80, 110, 137, 184, 229, 303, 375, 486, 602, 772, 951, 1202, 1478, 1853, 2267, 2817, 3432, 4236, 5142, 6300, 7620, 9284, 11185, 13553, 16273, 19625, 23478, 28187, 33613, 40192, 47778, 56904, 67443, 80051

Offset: 4

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=4 of A197126.

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]], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=4..60);

b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m == 4, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 4, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^4/(1-x^4)-x^5/(1-x^5))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (40*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A222705 Total number of parts of multiplicity 5 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 5, 9, 11, 17, 21, 31, 37, 53, 67, 90, 113, 151, 186, 246, 305, 392, 486, 620, 762, 962, 1181, 1473, 1802, 2235, 2716, 3345, 4056, 4956, 5990, 7283, 8759, 10598, 12709, 15297, 18283, 21917, 26099, 31165, 37009, 44014, 52113, 61776, 72918

Offset: 5

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=5 of A197126.

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]], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=5..60);

b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 5, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 5, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^5/(1-x^5)-x^6/(1-x^6))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (60*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A222706 Total number of parts of multiplicity 6 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 8, 10, 15, 18, 28, 33, 47, 58, 79, 97, 132, 161, 212, 262, 337, 414, 531, 648, 818, 1001, 1249, 1519, 1887, 2285, 2812, 3401, 4155, 5004, 6086, 7301, 8827, 10565, 12708, 15155, 18162, 21587, 25757, 30539, 36296, 42904, 50832, 59915

Offset: 6

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=6 of A197126.

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]], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=6..60);

b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 6, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 6, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^6/(1-x^6)-x^7/(1-x^7))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (84*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A222707 Total number of parts of multiplicity 7 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 17, 25, 30, 43, 53, 72, 88, 118, 145, 190, 234, 301, 370, 471, 575, 724, 884, 1102, 1339, 1657, 2007, 2465, 2975, 3630, 4369, 5301, 6355, 7672, 9171, 11018, 13126, 15706, 18655, 22231, 26329, 31260, 36920, 43688, 51454, 60693

Offset: 7

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=7 of A197126.

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]], 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, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 7, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 7, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^7/(1-x^7)-x^8/(1-x^8))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (112*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A222708 Total number of parts of multiplicity 8 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 8, 9, 13, 16, 24, 28, 40, 49, 67, 82, 110, 133, 176, 215, 276, 338, 432, 524, 661, 804, 1001, 1214, 1502, 1812, 2228, 2683, 3270, 3928, 4766, 5699, 6879, 8208, 9855, 11724, 14020, 16624, 19804, 23423, 27788, 32781, 38770, 45603, 53762

Offset: 8

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=8 of A197126.

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]], 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, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 8, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 8, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^8/(1-x^8)-x^9/(1-x^9))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (144*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

A222709 Total number of parts of multiplicity 9 in all partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 9, 13, 15, 23, 27, 38, 47, 63, 77, 104, 126, 165, 202, 259, 316, 403, 489, 614, 748, 929, 1125, 1391, 1676, 2055, 2475, 3012, 3613, 4379, 5233, 6306, 7521, 9018, 10717, 12805, 15171, 18050, 21337, 25288, 29806, 35221, 41400, 48760

Offset: 9

Alois P. Heinz, Feb 28 2013

nonn

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

Column k=9 of A197126.

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]], 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, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 9, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 9, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

G.f.: (x^9/(1-x^9)-x^10/(1-x^10))/Product_{j>0}(1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (180*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018

cp's OEIS Frontend

A116646 Number of doubletons in all partitions of n. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

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A117524 Total number of parts of multiplicity 3 in all partitions of n.

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A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

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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.

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A222704 Total number of parts of multiplicity 4 in all partitions of n.

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A222705 Total number of parts of multiplicity 5 in all partitions of n.

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A222706 Total number of parts of multiplicity 6 in all partitions of n.

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A222707 Total number of parts of multiplicity 7 in all partitions of n.

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