A187219 -id:A187219 - cp's OEIS frontend

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

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

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 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] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {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]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 20 2012

A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

Original entry on oeis.org

1, 4, 8, 1, 15, 4, 1, 21, 8, 4, 1, 1, 33, 15, 8, 4, 4, 1, 1, 41, 21, 15, 8, 8, 4, 4, 1, 1, 1, 1, 56, 33, 21, 15, 15, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 87, 56, 41, 33, 33, 21, 21, 15, 15, 15, 15, 8, 8, 8, 8

Offset: 1

Omar E. Pol, Jan 07 2021

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the tower described in A221529 whose largest side of the base is equal to n (see example). - Omar E. Pol, Jan 08 2022

			Triangle begins:
   1;
   4;
   8,  1;
  15,  4,  1;
  21,  8,  4,  1,  1;
  33, 15,  8,  4,  4,  1,  1;
  41, 21, 15,  8,  8,  4,  4, 1, 1, 1, 1;
  56, 33, 21, 15, 15,  8,  8, 4, 4, 4, 4, 1, 1, 1, 1;
  69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 9 the length of row 9 is A000041(9-1) = 22.
From _Omar E. Pol_, Jan 08 2022: (Start)
For n = 9 the lateral view and top view of the tower described in A221529 look like as shown below:
                        _
    22        1        | |
    21        1        | |
    20        1        | |
    19        1        | |
    18        1        | |
    17        1        | |
    16        1        |_|_
    15        4        |   |
    14        4        |   |
    13        4        |   |
    12        4        |_ _|_
    11        8        |   | |
    10        8        |   | |
     9        8        |   | |
     8        8        |_ _|_|_
     7       15        |     | |
     6       15        |_ _ _| |_
     5       21        |     |   |
     4       21        |_ _ _|_ _|_
     3       33        |_ _ _ _| | |_
     2       41        |_ _ _ _|_|_ _|_ _
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view
     k     T(9,k)        of the tower
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | |   |
                       |_ _|_| | | | |   |
                       |_ _|  _|_| | |   |
                       |_ _ _|    _|_|   |
                       |_ _ _|  _|    _ _|
                       |_ _ _ _|     |
                       |_ _ _ _|  _ _|
                       |         |
                       |_ _ _ _ _|
.
                           Top view
                         of the tower
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the tower, so T(9,1) = 69.
For n = 9 and k = 22 there is only one cubic cell in the level 22 (the top) of the tower, so T(9,22) = 1.
The volume of the tower (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000041 hence it's equal to A066186(9) = 270, equaling the sum of the 9th row of triangle. (End)

Row sums give A066186.
Row lengths give A000041.
The length of the m-th block in row n is A187219(m), m >= 1.
Cf. A350637 (analog for the stepped pyramid described in A245092).
Cf. A000203, A024916, A196020, A221529, A236104, A235791, A237270, A237271, A237593, A339278, A262626, A336811, A338156, A340035, A341149, A346533, A350333.

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; } \\ A336811
g(n) = sum(k=1, n, n\k*k); \\ A024916
row(n) = vector(f(n), k, g(T(n,k))); \\ Michel Marcus, Jan 22 2022

T(n,k) = A024916(A336811(n,k)).

T(n,k) = Sum_{j=1..n} A339278(j,k). - Omar E. Pol, Jan 08 2022

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281

Offset: 1

Omar E. Pol, Jan 25 2011

Question: Is there some connection with fractals?

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.

Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

New name from Omar E. Pol, Apr 26 2012

A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

Original entry on oeis.org

0, 0, 0, 2, 5, 16, 30, 63, 108, 189, 298, 483, 720, 1092, 1582, 2297, 3225, 4551, 6244, 8592, 11590, 15622, 20741, 27536, 36066, 47198, 61150, 79077, 101391, 129808, 164934, 209213, 263745, 331807, 415229, 518656, 644719, 799926, 988432, 1218979

Offset: 1

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182709. Total sum of emergent parts in all partitions of all numbers <= n.

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196039, A196930, A196931, A198381.

a(n) = A066186(n) - A196039(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)). - Vaclav Kotesovec, Jul 06 2019

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.

A207038 Partial sums of A207034.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 18, 22, 26, 31, 35, 40, 45, 51, 55, 60, 65, 71, 77, 83, 90, 95, 101, 107, 114, 120, 127, 134, 142, 147, 153, 159, 166, 173, 180, 188, 195, 203, 211, 219, 228, 234, 241, 248, 256, 263, 271, 279, 288, 296, 304, 313, 322, 331, 341

Offset: 1

Omar E. Pol, Feb 21 2012

			Written as a triangle:
0;
1;
3;
5,8;
11,15;
18,22,26,31;
35,40,45,51;
55,60,65,71,77,83,90;
95,101,107,114,120,127,134,142;
147,153,159,166,173,180,188,195,203,211,219,228;
234,241,248,256,263,271,279,288,296,304,313,322,331,341;

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

Row n has length A187219(n). Right border gives A196087.

Cf. A006128, A135010, A138121, A194548, A207035.

A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 3, 2, 1, 1, 4, 4, 4, 3, 2, 1, 1, 7, 7, 6, 5, 3, 2, 1, 1, 8, 8, 8, 6, 5, 3, 2, 1, 1, 12, 12, 11, 10, 7, 5, 3, 2, 1, 1, 14, 14, 14, 12, 10, 7, 5, 3, 2, 1, 1, 21, 21, 20, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 10 2012

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

			Illustration of initial terms. First six rows of triangle as numbers of parts in the columns from the last sections of the first six natural numbers:
.                                       6
.                                       3 3
.                                       4 2
.                                       2 2 2
.                           5             1
.                           3 2             1
.                 4           1             1
.                 2 2           1             1
.         3         1           1             1
.   2       1         1           1             1
1     1       1         1           1             1
---------------------------------------------------
1,  1,1,  1,1,1,  2,2,1,1,  2,2,2,1,1,  4,4,3,2,1,1
...
Triangle begins:
1;
1,   1;
1,   1,  1;
2,   2,  1,  1;
2,   2,  2,  1,  1;
4,   4,  3,  2,  1,  1;
4,   4,  4,  3,  2,  1,  1;
7,   7,  6,  5,  3,  2,  1,  1;
8,   8,  8,  6,  5,  3,  2,  1,  1;
12, 12, 11, 10,  7,  5,  3,  2,  1,  1;
14, 14, 14, 12, 10,  7,  5,  3,  2,  1,  1;
21, 21, 20, 18, 14, 11,  7,  5,  3,  2,  1,  1;

Column 1 is A187219. Row sums give A138137. Reversed rows converge to A000041.

Cf. A002865, A058399, A135010, A138135, A181187, A207031, A207380.

A194795 Imbalance of the number of partitions of n.

Original entry on oeis.org

0, -1, 0, -2, 0, -4, 0, -7, 1, -11, 3, -18, 6, -28, 13, -42, 24, -64, 41, -96, 69, -141, 112, -208, 175, -303, 271, -437, 410, -629, 609, -898, 896, -1271, 1302, -1792, 1868, -2510, 2660, -3493, 3752, -4839, 5248, -6666, 7293, -9131, 10065, -12454

Offset: 1

Omar E. Pol, Feb 02 2012

sign

Consider the three-dimensional structure of the shell model of partitions version "tree". Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns. Note that every column contains exactly the same parts, the same as a periodic table (see example). For more information see A135010.

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
The number of partitions with parts on the left hand side is equal to 7 and the number of partitions with parts on the right hand side is equal to 3, so a(6) = -7+3 = -4. On the other hand; for n = 6 the first n terms of A002865 (with positive indices) are 0, 1, 1, 2, 2, 4 therefore a(6) = 0-1+1-2+2-4 = -4.

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

Cf. A000041, A002865, A135010, A182703, A187219, A194796, A194797, A194805, A194809.

with(combinat):
a:= proc(n) option remember;
      (-1)^n *(numbpart(n-1)-numbpart(n)) +`if`(n>1, a(n-1), 0)
    end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

a[n_] := a[n] = (-1)^n*(PartitionsP[n-1]-PartitionsP[n]) + If[n>1, a[n-1], 0]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
nmax = 60; Rest[CoefficientList[Series[x/(1-x) - (1+x)/(1-x) * Product[1/((1 + x^(2*k-1))*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)
nmax = 60; Rest[CoefficientList[Series[-x/(1+x) - (1-x)/(1+x) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2015 *)

a(n) = Sum_{k=1..n} (-1)^(k-1)*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = Sum_{k=1..n} (-1)^(k-1)*A002865(k).
a(n) = (-1)^(n+1) * (A240690(n+1) - A240690(n)) - 1. - Vaclav Kotesovec, Nov 11 2015
a(n) ~ (-1)^(n+1) * Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Nov 11 2015

A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 2, 2, 4, 2, 3, 6, 3, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 10, 5

Offset: 1

Omar E. Pol, Apr 18 2012

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

			Written as a triangle begins:
1;
2;
3;
2;
4,2;
3;
5,2,
2;
4,2;
3;
6,3,2,2;
3;
5,2;
4;
7,3,2,2;

Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285. Records give A000027. The n-th record is T(A000041(n),1).

Cf. A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A198381, A206437, A210941.

A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

Original entry on oeis.org

4, 6, 8, 12, 16, 24, 32, 46, 62, 86, 114, 156, 204, 272, 354, 464, 596, 772, 982, 1256, 1586, 2006, 2512, 3152, 3918, 4874, 6022, 7438, 9132, 11210, 13686, 16700, 20288, 24622, 29768, 35956, 43276, 52032, 62372, 74678, 89168, 106350

Offset: 1

Omar E. Pol, Oct 29 2012

nonn

On the infinite square grid the diagram of regions of the set of partitions of n is represented by a rectangle with base = n and height = A000041(n). The rectangle contains n shells. Each shell contains regions. Each row of a region is a part. Each part of size k contains k cells. The number of regions equals the number of partitions of n (see illustrations in the links section). For a minimalist version see A139582. For the definition of "region of n" see A206437.

Omar E. Pol, Illustration of initial terms of A211026 and of A066186
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A052810, A135010, A139582, A141285, A186412, A186114, A187219, A193870, A194446, A194447, A206437, A211009

a(n) = 2*A000041(n) + 2 = 2*A052810(n) = A139582(n) + 2.

a(18) corrected by Georg Fischer, Apr 11 2024

cp's OEIS Frontend

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

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A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

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A182727 Sum of largest parts of the shell model of partitions with n regions.

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A196025 Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.

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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.

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A207038 Partial sums of A207034.

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A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

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A194795 Imbalance of the number of partitions of n.

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A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

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