A213177 -id:A213177 - cp's OEIS frontend

A015723 Number of parts in all partitions of n into distinct parts.

1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691

Offset: 1

Clark Kimberling

nonn

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Arnold Knopfmacher, and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See s(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A006128, A008289, A079499, A067619, A186545.
Column k=1 of: A210485, A213177, A327622.
Row lengths of A325537.
Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013

nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i,{i,1,nn}],y]/.y->1,{x,0,nn}],x]]  (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[nVaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,50}] (* Gus Wiseman, May 09 2019 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
   Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

N=66;  q='q+O('q^N); gf=sum(n=0,N, n*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf) /* Joerg Arndt, Oct 20 2012 */

G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - Vladeta Jovovic, Nov 26 2002
G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014
a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016
G.f.: -(-1; x)inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018

Extended and corrected by Naohiro Nomoto, Feb 24 2002

A091602 Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 3, 2, 1, 0, 1, 5, 4, 3, 1, 1, 0, 1, 6, 7, 3, 3, 1, 1, 0, 1, 8, 8, 6, 3, 2, 1, 1, 0, 1, 10, 12, 7, 5, 3, 2, 1, 1, 0, 1, 12, 15, 11, 6, 5, 2, 2, 1, 1, 0, 1, 15, 21, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1, 18, 26, 20, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1, 22, 35, 25, 18, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 1

Christian G. Bower, Jan 23 2004

From Gary W. Adamson, Mar 13 2010: (Start)
The triangle by rows = finite differences starting from the top, of an array in which row 1 = p(x)/p(x^2), row 2 = p(x)/p(x^3), ... row k = p(x)/p(x^k); such that p(x) = polcoeff A000041: (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...)
Note that p(x)/p(x^2) = polcoeff A000009: (1 + x + x^2 + 2x^3 + 2x^4 + ...).
Refer to the example. (End)

			Triangle starts:
   1:  1;
   2:  1,  1;
   3:  2,  0,  1;
   4:  2,  2,  0,  1;
   5:  3,  2,  1,  0,  1;
   6:  4,  3,  2,  1,  0,  1;
   7:  5,  4,  3,  1,  1,  0,  1;
   8:  6,  7,  3,  3,  1,  1,  0,  1;
   9:  8,  8,  6,  3,  2,  1,  1,  0,  1;
  10: 10, 12,  7,  5,  3,  2,  1,  1,  0,  1;
  11: 12, 15, 11,  6,  5,  2,  2,  1,  1,  0,  1;
  12: 15, 21, 14, 10,  5,  5,  2,  2,  1,  1,  0,  1;
  13: 18, 26, 20, 12,  9,  5,  4,  2,  2,  1,  1,  0,  1;
  14: 22, 35, 25, 18, 11,  8,  5,  4,  2,  2,  1,  1,  0,  1;
  ...
In the partition 5+2+2+2+1+1, 2 is repeated 3 times, no part is repeated more than 3 times.
From _Gary W. Adamson_, Mar 13 2010: (Start)
First few rows of the array =
  ...
  1, 1, 1, 2, 2, 3,  4,  5,  6,  8, 10, ... = p(x)/p(x^2) = A000009
  1, 1, 2, 2, 4, 5,  7,  9, 13, 16, 22, ... = p(x)/p(x^3)
  1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, ... = p(x)/p(x^4)
  1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ... = p(x)/p(x^5)
  1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ... = p(x)/p(x^6)
  ...
Finally, taking finite differences from the top and deleting the first "1", we obtain triangle A091602 with row sums = A000041 starting with offset 1:
  1;
  1, 1;
  2, 0, 1;
  2, 2, 0, 1;
  3, 2, 1, 0, 1;
  4, 3, 2, 1, 0, 1;
  ...
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Row sums: A000041. Inverse: A091603. Square: A091604.
Columns 1-6: A000009, A091605-A091609. Convergent of columns: A002865.
Cf. A000009. - Gary W. Adamson, Mar 13 2010
T(2n,n) gives: A232697.
Cf. A213177, A264397.

g:=sum(t^k*(product((1-x^((k+1)*j))/(1-x^j),j=1..50)-product((1-x^(k*j))/(1-x^j),j=1..50)),k=1..50): gser:=simplify(series(g,x=0,20)): for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 1 to 13 do seq(coeff(P[n],t^j),j=1..n) od;
# yields sequence in triangular form - Emeric Deutsch, Mar 30 2006
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, 1, k) -`if`(k=0, 0, b(n, 1, k-1)):
seq(seq(T(n, k), k=1..n), n=1..20);
# Alois P. Heinz, Nov 27 2013

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; t[n_, k_] := b[n, 1, k] - If[k == 0, 0, b[n, 1, k-1]]; Table[t[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz's second Maple program *)

G.f.: G = G(t,x) = sum(k>=1, t^k*(prod(j>=1, (1-x^((k+1)*j))/(1-x^j) ) -prod(j>=1, (1-x^(k*j))/(1-x^j) ) ) ). - Emeric Deutsch, Mar 30 2006

Sum_{k=1..n} k * T(n,k) = A264397(n). - Alois P. Heinz, Nov 20 2015

A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192

Offset: 0

Alois P. Heinz, Jan 23 2013

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.

For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

			T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
  0;
  0,  1;
  0,  1,  3;
  0,  3,  3,  6;
  0,  3,  8,  8, 12;
  0,  5, 11, 15, 15, 20;
  0,  8, 17, 24, 29, 29, 35;
  0, 10, 23, 36, 41, 47, 47, 54;
  0, 13, 36, 50, 65, 71, 78, 78, 86;
  ...

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

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.
Main diagonal gives A006128.
T(2n,n) gives A364245.
Cf. A213177, A286653.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A213177(n,i).

A320372 Number of parts in all partitions of n with largest multiplicity two.

Original entry on oeis.org

2, 0, 5, 6, 9, 13, 23, 30, 44, 58, 85, 108, 149, 191, 258, 326, 425, 532, 688, 852, 1082, 1331, 1670, 2042, 2531, 3068, 3771, 4554, 5543, 6653, 8051, 9607, 11543, 13722, 16377, 19390, 23023, 27132, 32073, 37660, 44303, 51834, 60744, 70813, 82666, 96082, 111759

Offset: 2

Alois P. Heinz, Oct 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..5000

Column k=2 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(2):
seq(a(n), n=2..60);

b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[ Function[l, {0, l[[1]]*j} + l][b[n - i j, i-1, k]], {j, 0, Min[n/i, k]}]]];
a[n_] := With[{k = 2}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
a /@ Range[2, 60] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320373 Number of parts in all partitions of n with largest multiplicity three.

Original entry on oeis.org

3, 0, 4, 7, 13, 14, 27, 33, 55, 72, 107, 137, 196, 250, 344, 442, 588, 750, 982, 1234, 1591, 1992, 2523, 3135, 3944, 4857, 6035, 7408, 9121, 11109, 13599, 16465, 20004, 24122, 29112, 34927, 41952, 50078, 59836, 71169, 84625, 100219, 118716, 140061, 165225

Offset: 3

Alois P. Heinz, Oct 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Column k=3 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(3):
seq(a(n), n=3..60);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0},        Sum[Function[l, {0, l[[1]] j} + l][b[n - i j, i - 1, k]], {j, 0, Min[n/i, k]}]]];
a[n_] := With[{k = 3}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320374 Number of parts in all partitions of n with largest multiplicity four.

Original entry on oeis.org

4, 0, 5, 5, 15, 16, 30, 36, 60, 75, 116, 149, 217, 273, 386, 491, 664, 839, 1116, 1399, 1829, 2292, 2937, 3656, 4638, 5729, 7187, 8840, 10984, 13430, 16558, 20138, 24657, 29846, 36288, 43736, 52880, 63430, 76289, 91159, 109106, 129841, 154724, 183452, 217727

Offset: 4

Alois P. Heinz, Oct 11 2018

nonn

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

Column k=4 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(4):
seq(a(n), n=4..50);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0},        Sum[Function[l, {0, l[[1]] j} + l][b[n - i j, i - 1, k]], {j, 0, Min[n/i, k]}]]];
a[n_] := With[{k = 4}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

a(n) ~ 3^(1/4) * log(5) * exp(2*Pi*sqrt(2*n/15)) / (2^(5/4) * 5^(1/4) * Pi * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320375 Number of parts in all partitions of n with largest multiplicity five.

Original entry on oeis.org

5, 0, 6, 6, 13, 18, 34, 35, 66, 82, 120, 154, 230, 286, 408, 514, 699, 886, 1189, 1485, 1949, 2441, 3136, 3906, 4980, 6159, 7757, 9555, 11908, 14600, 18062, 22000, 27028, 32804, 39996, 48327, 58614, 70489, 85036, 101876, 122284, 145943, 174419, 207354, 246804

Offset: 5

Alois P. Heinz, Oct 11 2018

nonn

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

Column k=5 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(5):
seq(a(n), n=5..50);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0},        Sum[Function[l, {0, l[[1]] j} + l][b[n - i j, i - 1, k]], {j, 0, Min[n/i, k]}]]];
a[n_] := With[{k = 5}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
a /@ Range[5, 50] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

a(n) ~ log(6) * exp(Pi*sqrt(5*n)/3) / (2 * Pi * 5^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320376 Number of parts in all partitions of n with largest multiplicity six.

Original entry on oeis.org

6, 0, 7, 7, 15, 15, 38, 39, 67, 84, 131, 160, 242, 293, 421, 537, 730, 912, 1234, 1533, 2021, 2525, 3269, 4053, 5185, 6394, 8080, 9948, 12425, 15219, 18893, 23006, 28319, 34379, 42024, 50788, 61736, 74279, 89795, 107674, 129483, 154660, 185221, 220424, 262820

Offset: 6

Alois P. Heinz, Oct 11 2018

nonn

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

Column k=6 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(6):
seq(a(n), n=6..50);

a(n) ~ log(7) * exp(2*Pi*sqrt(n/7)) / (2 * Pi * 7^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320377 Number of parts in all partitions of n with largest multiplicity seven.

Original entry on oeis.org

7, 0, 8, 8, 17, 17, 36, 43, 74, 85, 136, 167, 254, 308, 437, 550, 754, 944, 1270, 1581, 2088, 2588, 3356, 4160, 5326, 6568, 8306, 10217, 12776, 15634, 19413, 23644, 29134, 35360, 43250, 52285, 63599, 76547, 92608, 111079, 133705, 159774, 191477, 228012, 272104

Offset: 7

Alois P. Heinz, Oct 11 2018

nonn

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

Column k=7 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(7):
seq(a(n), n=7..50);

a(n) ~ 3^(5/4) * log(2) * exp(Pi*sqrt(7*n/3)/2) / (2^(3/2) * 7^(1/4) * Pi * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

A320378 Number of parts in all partitions of n with largest multiplicity eight.

Original entry on oeis.org

8, 0, 9, 9, 19, 19, 40, 40, 81, 93, 140, 173, 267, 315, 461, 572, 779, 970, 1314, 1619, 2152, 2662, 3442, 4265, 5468, 6706, 8502, 10434, 13065, 15969, 19853, 24130, 29769, 36093, 44166, 53357, 64957, 78120, 94585, 113406, 136564, 163163, 195657, 232927, 278157

Offset: 8

Alois P. Heinz, Oct 11 2018

nonn

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

Column k=8 of A213177.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l->l+[0, l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(8):
seq(a(n), n=8..50);

a(n) ~ log(3) * exp(4*Pi*sqrt(n/3)/3) / (Pi * sqrt(2) * 3^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018

cp's OEIS Frontend

A015723 Number of parts in all partitions of n into distinct parts.

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A091602 Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times.

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A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A320372 Number of parts in all partitions of n with largest multiplicity two.

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A320373 Number of parts in all partitions of n with largest multiplicity three.

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A320374 Number of parts in all partitions of n with largest multiplicity four.

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A320375 Number of parts in all partitions of n with largest multiplicity five.

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A320376 Number of parts in all partitions of n with largest multiplicity six.

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