A096541 -id:A096541 - cp's OEIS frontend

A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n.

1, 3, 1, 6, 2, 1, 12, 5, 2, 1, 20, 8, 4, 2, 1, 35, 16, 8, 4, 2, 1, 54, 24, 13, 7, 4, 2, 1, 86, 41, 22, 13, 7, 4, 2, 1, 128, 61, 35, 20, 12, 7, 4, 2, 1, 192, 95, 54, 33, 20, 12, 7, 4, 2, 1, 275, 136, 80, 49, 31, 19, 12, 7, 4, 2, 1, 399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1, 556, 284

Offset: 1

Wouter Meeussen, Oct 09 2010

For the connection with A066897 and A066898 see A206563. - Omar E. Pol, Feb 13 2012
T(n,k) is also the total number of parts >= k in all partitions of n. - Omar E. Pol, Feb 14 2012
The first differences of row n together with 1 give the row n of triangle A066633. - Omar E. Pol, Feb 26 2012
We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1. It appears that the sum of the k-th ranks of all partitions of n is equal to zero. - Omar E. Pol, Mar 04 2012
T(n,k) is also the total number of divisors >= k of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

			From _Omar E. Pol_, Feb 13 2012: (Start)
Illustration of initial terms. First five rows of triangle as sums of columns from the partitions of the first five positive integers:
.
.                            5
.                            3+2
.                  4         4+1
.                  2+2       2+2+1
.          3       3+1       3+1+1
.     2    2+1     2+1+1     2+1+1+1
.  1  1+1  1+1+1   1+1+1+1   1+1+1+1+1
. -------------------------------------
.  1, 3,1, 6,2,1, 12,5,2,1, 20,8,4,2,1 --> This triangle
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|
.  1, 2,1, 4,1,1,  7,3,1,1, 12,4,2,1,1 --> A066633
.
For more information see A207031 and A206563.
...
Triangle begins:
    1;
    3,   1;
    6,   2,   1;
   12,   5,   2,  1;
   20,   8,   4,  2,  1;
   35,  16,   8,  4,  2,  1;
   54,  24,  13,  7,  4,  2,  1;
   86,  41,  22, 13,  7,  4,  2,  1;
  128,  61,  35, 20, 12,  7,  4,  2, 1;
  192,  95,  54, 33, 20, 12,  7,  4, 2, 1;
  275, 136,  80, 49, 31, 19, 12,  7, 4, 2, 1;
  399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1;
(End)

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

Row sums are A066186. First column is A006128. Reverse of each row converges to A000070.
Columns 2-3: A096541, A207033. - Omar E. Pol, Feb 18 2012
T(2n,n) gives A216053(n+1).
Cf. A206283.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p(p(f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n) local j, l, r, t;
      l, r, t:= b(n, n), 1, 1;
      for j from n to 2 by -1 do t:= t+l[j]; r:=r, t od;
      seq([r][1+n-j], j=1..n)
    end:
seq(T(n), n=1..14); # Alois P. Heinz, Apr 05 2012

Table[Plus @@ (PadRight[ #,n]& /@ IntegerPartitions[n]),{n,16}]
(* Second program: *)
T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; Table[Table[T[n, k], {k, n, 1, -1}] // Accumulate // Reverse, {n, 1, 16}] // Flatten (* Jean-François Alcover, Oct 10 2015, after Omar E. Pol *)

T(n,k) = Sum_{j=1..n} A207031(j,k). - Omar E. Pol, May 02 2012

Better definition from Omar E. Pol, Feb 13 2012

A138135 Number of parts > 1 in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 8, 17, 20, 34, 41, 68, 80, 123, 153, 219, 271, 382, 469, 642, 795, 1055, 1305, 1713, 2102, 2713, 3336, 4241, 5190, 6545, 7968, 9950, 12090, 14953, 18104, 22255, 26821, 32752, 39371, 47774, 57220, 69104

Offset: 1

Omar E. Pol, Mar 30 2008

nonn

Also first differences of A096541. For more information see A135010.

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

Zero together with the column k=2 of A207031.

Cf. A000041, A006128, A096541, A135010, A138121, A138137.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+`if`(i>1, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]-b(n-1, n-1)[2]:
seq (a(n), n=1..60); # Alois P. Heinz, Apr 04 2012

a[n_] := DivisorSigma[0, n] - 1 + Sum[(DivisorSigma[0, k] - 1)*(PartitionsP[n - k] - PartitionsP[n - k - 1]), {k, 1, n - 1}]; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 14 2013, from 1st formula *)
Table[Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 42}]  (* Robert Price, May 01 2020 *)

a(n)=numdiv(n)-1+sum(k=1,n-1,(numdiv(k)-1)*(numbpart(n-k) - numbpart(n-k-1))) \\ Charles R Greathouse IV, Jan 14 2013

a(n) = A096541(n)-A096541(n-1) = A138137(n)-A000041(n-1) = A006128(n)-A006128(n-1)-A000041(n-1).
a(n) ~ exp(Pi*sqrt(2*n/3))*(2*gamma - 2 + log(6*n/Pi^2))/(8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 24 2016
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k) / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n.

Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.

a(n) = A096541(n) - A000065(n) = 1 + A096541(n) - A000041(n) = 1 + A006128(n) - A000070(n).

a(n) = A006128(n) - A026905(n), n >= 1.

A194552 Sum of all parts > 1 of all partitions of n.

Original entry on oeis.org

0, 2, 5, 13, 23, 47, 75, 131, 203, 323, 477, 729, 1041, 1517, 2132, 3012, 4134, 5718, 7713, 10453, 13918, 18538, 24357, 32037, 41612, 54040, 69538, 89362, 113925, 145095, 183473, 231697, 290899, 364577, 454632, 566016, 701436, 867800, 1069430, 1315550, 1612595

Offset: 1

Omar E. Pol, Dec 11 2011

nonn

Also the total number of missing parts in the partitions of n. A missing part of a partition of n is any number from 1 to n not occurring as a part. For example for n = 3, 1,2 are missing from 3; 3 is missing from 2+1, and 2,3 are missing from 1+1+1, for a total of a(3) = 5. - George Beck, Oct 23 2014

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

Partial sums of A138880.

Cf. A096541, A135010, A138121, A138135.

b:= proc(n, i) option remember; local h, t;
      if n<0 or i<1 then [0, 0]
    elif n=0 or i=1 then [1, 0]
    else h:= b(n, i-1); t:= b(n-i, i);
         [h[1]+t[1], h[2]+t[2] +t[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50); # Alois P. Heinz, Dec 14 2011

a[n_] := n PartitionsP[n] -Total@Table[PartitionsP[k], {k, 0, n - 1}]; a /@ Range[40] (* George Beck, Oct 23 2014 *)

a(n) = A066186(n) - A000070(n-1).
a(n) = n * A000041(n) - A000070(n-1). - George Beck, Oct 24 2014
G.f.: (x/(1 - x)) * (d/dx) Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Mar 06 2021

A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 61, 107, 176, 284, 442, 676, 1007, 1483, 2140, 3055, 4299, 5993, 8255, 11284, 15272, 20529, 27373, 36274, 47735, 62484, 81293, 105251, 135555, 173818, 221836, 282003, 356980, 450256, 565765, 708537, 884296, 1100287, 1364736, 1687952, 2081724

Offset: 0

Alois P. Heinz, Nov 18 2020

nonn

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

Cf. A096541, A339006, A339312.

Essentially partial sums of A093694.

b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],
     `if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))
    end:
a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 25 2022

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 17, 23, 33, 47, 71, 92, 129, 169, 235, 299, 408, 525, 691, 885, 1147, 1427, 1832, 2312, 2878, 3635, 4519, 5631, 7002, 8637, 10514, 13055, 15864, 19396, 23530, 28702, 34746, 42210, 50671, 61224, 73506, 88394, 105447, 126398, 150588, 179075

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

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

Cf. A003989, A050873, A096541, A339006, A339011, A339394.

b:= proc(n, i, p, d) option remember; `if`(n=0, igcd(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, GCD[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A194796 Imbalance of the number of parts of all partitions of n.

Original entry on oeis.org

0, -1, 0, -3, 0, -8, 0, -17, 3, -31, 10, -58, 22, -101, 52, -167, 104, -278, 191, -451, 344, -711, 594, -1119, 983, -1730, 1606, -2635, 2555, -3990, 3978, -5972, 6118, -8835, 9269, -12986, 13835, -18917, 20454, -27320, 29900, -39204, 43268, -55846, 62112

Offset: 1

Omar E. Pol, Feb 01 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see the illustration in A194795). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. For more information see A135010.

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

Cf. A006128, A096541, A135010, A138121, A138135, A138137, A197595, A194797, A194809.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= proc(n) option remember;
      (-1)^n*(b(n-1, n-1)[2]-b(n, n)[2])+`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 04 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := a[n] = (-1)^n*(b[n-1, n-1][[2]] - b[n, n][[2]]) + If[n == 1, 0, a[n-1]]; Table [a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

vector(50, n, sum(k=1, n, (-1)^(k-1)*(numdiv(k)-1+sum(j=1, k-1, (numdiv(j)-1)*(numbpart(k-j)-numbpart(k-j-1)))))) \\ Altug Alkan, Nov 11 2015

a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).

More terms from Alois P. Heinz, Apr 04 2012

A207033 Total number of parts >= 3 in all partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 80, 121, 172, 247, 347, 484, 661, 906, 1215, 1632, 2162, 2855, 3730, 4871, 6290, 8111, 10381, 13252, 16802, 21269, 26750, 33583, 41948, 52277, 64862, 80326, 99055, 121922, 149541, 183052, 223350, 272038, 330343, 400450, 484154

Offset: 1

Omar E. Pol, Feb 18 2012

nonn

			a(4) = 2, because 2 parts have size >= 3 in all partitions of 4: [1,1,1,1], [1,1,2], [2,2], [1,3], [4].

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

Column 3 of A181187.

Cf. A006128, A096541, A206563.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +`if`(i>2, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i > n, b[n, i - 1], True, f = b[n, i - 1]; g = b[n - i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i > 2, g[[1]], 0]}]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 18 2012

cp's OEIS Frontend

A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A138135 Number of parts > 1 in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

Views

Author

Keywords

Comments

Crossrefs

Formula

A194552 Sum of all parts > 1 of all partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A194796 Imbalance of the number of parts of all partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica