A135010 -id:A135010 - cp's OEIS frontend

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

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

A182712 Number of 2's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Omar E. Pol, Nov 28 2010

Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - Omar E. Pol, Nov 13 2011
Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - Omar E. Pol, Dec 01 2013

			a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
From _Omar E. Pol_, Oct 27 2012: (Start)
----------------------------------
Last section               Number
of the set of                of
partitions of 6             2's
----------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
.   1 ...................... 0
.       1 .................. 0
.       1 .................. 0
.           1 .............. 0
.           1 .............. 0
.               1 .......... 0
.                   1 ...... 0
---------------------------------
.   8 - 4 =                  4
.
In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
(End)

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

Column 2 of A194812.

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.

Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* Robert Price, May 15 2020 *)

A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # Omar E. Pol, Nov 13 2011

It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - Omar E. Pol, Feb 04 2012
G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jun 02 2018

A182742 Table of partitions that do not contain 1 as a part for even integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 30 2010, Dec 01 2010, Dec 04 2010

This array read by antidiagonals is the main table of the shell model of partitions for even integers. Here the last sections of all even numbers are superimposed as shells of an onion. In this way many bits of information are saved.
The table is the head of the last section of partitions of an even integer when it tends to be infinite. Row n lists the parts of the n-th partition that do not contains 1 as a part.
The shell model of partitions uses this table during the filling mechanism of the head of the last section of the next even integer k. For example, in a mechanical version, the head of the last section (as a mirror) pivoting from vertical to horizontal position. Then a copy of the partitions of the integer k, listed in this table, is transmitted (or reflected) at the head (or mirror) of the last section. Finally the head (or mirror) pivots back to return to its original vertical position. And so on for all even integers.
In another version, simply a copy of the partitions of the integer k, listed in the table, are placed above the partitions of the last odd number placed in the vertical plane structure.
It appears this table is useful to know the structure of the partitions of all even integers. The same applies for odd numbers in the table of A182743. Furthermore, both tables can be unified in a three-dimensional shell model.

			Array begins:
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 2, 2, 2, 2, 2, 2, 2, 2,
6, 2, 2, 2, 2, 2, 2, 2, 2,
5, 3, 2, 2, 2, 2, 2, 2,
4, 4, 2, 2, 2, 2, 2,
8, 2, 2, 2, 2, 2,
4, 3, 3, 2, 2,
7, 3, 2, 2,
6, 4, 2,
5, 5,
10,

Cf. A135010, A138121, A141285, A182730, A182736, A182740, A182743, A182746.

Column 1 give A182732. Column 2 give A182744.

A182743 Table of the partitions that do not contain 1 as a part for odd integers.

Original entry on oeis.org

3, 5, 2, 4, 2, 2, 7, 3, 2, 2, 3, 2, 2, 2, 2, 6, 3, 2, 2, 2, 2, 5, 3, 3, 2, 2, 2, 2, 9, 4, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Nov 30 2010, Dec 01 2010

The same idea as A182742 but for odd integers.

			Array begins:
3,2,2,2,2,2,2,2,2,2,2,
5,2,2,2,2,2,2,2,2,2,
4,3,2,2,2,2,2,2,2,2,
7,2,2,2,2,2,2,2,2,
3,3,3,2,2,2,2,2,2,2,
6,3,2,2,2,2,2,2,2,
5,4,2,2,2,2,2,2,2,
9,2,2,2,2,2,2,2,
5,3,3,2,2,2,2,2,2,
4,4,3,2,2,2,2,2,2,
8,3,2,2,2,2,2,2,
7,4,2,2,2,2,2,2,
6,5,2,2,2,2,2,2,
11,2,2,2,2,2,2,
4,3,3,3,2,2,2,2,2,
7,3,3,2,2,2,2,2,
6,4,3,2,2,2,2,2,
5,5,3,2,2,2,2,2,
10,3,2,2,2,2,2,
5,4,4,2,2,2,2,2,
9,4,2,2,2,2,2,
8,5,2,2,2,2,2,
7,6,2,2,2,2,2,

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A135010, A138121, A141285, A182731, A182742.

Column 1 give A182733. Column 2 give A182745.

cmpL := proc(a,b) local i ; for i from 1 to min(nops(a),nops(b)) do if op(i,a) < op(i,b) then return -1 ; elif op(i,a) > op(i,b) then return 1 ; end if; end do; if nops(a) > nops(b) then return 1; elif nops(a) < nops(b) then return -1; else return 0; end if; end proc:
pShellMin := proc(p) local idx,j; idx := 1 ; for j from 2 to nops(p) do if cmpL( op(j,p),op(idx,p)) < 0 then idx := j; end if; end do; return idx ; end proc:
A141285rowf := proc(n) local p; if n <= 1 then [n] ; else psort := [] ; p := combinat[partition](n) ; while nops(p) > 0 do m := pShellMin(p) ; mmi := min(op(op(m,p))) ; if mmi > 1 then mma := max(op(op(m,p))) ; psort := [op(psort),sort(op(m,p),`>`)] ; end if; p := subsop(m=NULL,p) ; end do: psort ; end if; end proc:
for n from 1 to 17 by 2 do shl := A141285rowf(n) ; for r in shl do for k in r do printf("%d,",k) ; end do: printf("\n") ; end do: printf("\n") ; end do: # R. J. Mathar, Dec 03 2010

A194812 Square array read by antidiagonals: T(n,k) = number of parts of size k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 0, 0, 0, 5, 2, 1, 0, 0, 7, 1, 0, 0, 0, 0, 11, 4, 1, 1, 0, 0, 0, 15, 3, 2, 0, 0, 0, 0, 0, 22, 8, 2, 1, 1, 0, 0, 0, 0, 30, 7, 3, 1, 0, 0, 0, 0, 0, 0, 42, 15, 6, 3, 1, 1, 0, 0, 0, 0, 0, 56, 15, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 77, 27, 10

Offset: 1

Omar E. Pol, Feb 04 2012

It appears that in the column k, starting in row n, the sum of k successive terms is equal to A000041(n-1).

			Array begins:
.  1,  0,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  1,  1,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  2,  0,  1,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  3,  2,  0,  1, 0, 0, 0, 0, 0, 0, 0, 0,...
.  5,  1,  1,  0, 1, 0, 0, 0, 0, 0, 0, 0,...
.  7,  4,  2,  1, 0, 1, 0, 0, 0, 0, 0, 0,...
. 11,  3,  2,  1, 1, 0, 1, 0, 0, 0, 0, 0,...
. 15,  8,  3,  3, 1, 1, 0, 1, 0, 0, 0, 0,...
. 22,  7,  6,  2, 2, 1, 1, 0, 1, 0, 0, 0,...
. 30, 15,  6,  5, 3, 2, 1, 1, 0, 1, 0, 0,...
. 42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1, 0,...
. 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1,...
...
For n = 7, from the conjecture we have that p(n-1) = p(6) = 11 = 3+8 = 2+3+6 = 1+3+2+5 = 1+1+2+3+4 = 0+1+1+2+2+5, etc. where p(n) = A000041(n).

Columns 1-4: A000041, A182712, A182713, A182714. Main triangle: A182703.

Cf. A066633, A135010, A138121, A138137.

It appears that A000041(n) = Sum_{j=1..k} T(n+j,k), n >= 0, k >= 1.

A139582 Twice partition numbers.

Original entry on oeis.org

2, 2, 4, 6, 10, 14, 22, 30, 44, 60, 84, 112, 154, 202, 270, 352, 462, 594, 770, 980, 1254, 1584, 2004, 2510, 3150, 3916, 4872, 6020, 7436, 9130, 11208, 13684, 16698, 20286, 24620, 29766, 35954, 43274, 52030, 62370, 74676, 89166, 106348, 126522, 150350, 178268, 211116, 249508, 294546, 347050, 408452

Offset: 0

Omar E. Pol, May 14 2008

Except for the first term the number of segments needed to draw (on the infinite square grid) a minimalist diagram of regions and partitions of n. Therefore A000041(n) is also the number of pairs of orthogonal segments (L-shaped) in the same diagram (See links section). For the definition of "regions of n" see A206437. - Omar E. Pol, Oct 29 2012

			The number of partitions of 6 is 11, then a(6) = 2*11 = 22.

V. Modrak, D. Marton, A framework for generating and complexity assessment of assembly supply chains, in Nonlinear Science and Complexity (NSC), 2012 IEEE 4th International Conference on, Date of Conference: 6-11 Aug. 2012; Digital Object Identifier: 10.1109/NSC.2012.6304712. - From _N. J. A. Sloane_, Dec 27 2012
V. Modrak, D. Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299; doi:10.3390/e15104285.
V. Modrak, D. Marton, Approaches to Defining and Measuring Assembly Supply Chain Complexity, Discontinuity and Complexity in Nonlinear Physical Systems, Vol. 6, 2014, pp. 192-213.
V. Modrak, D. Marton, Configuration complexity assessment of convergent supply chain systems, International Journal of General Systems, Volume 43, Issue 5, 2014.
Omar E. Pol, Illustration of initial terms of A139582 (n>=1) and of A066186

Cf. A000041, A135010, A206437, A211026.

Array[2 PartitionsP@# &, 50, 0] (* Robert G. Wilson v, Feb 11 2018 *)

a(n) = 2*numbpart(n); \\ Michel Marcus, Feb 12 2018

a(n) = 2*A000041(n).

More terms from Omar E. Pol, Feb 11 2018

A144300 Number of partitions of n minus number of divisors of n.

Original entry on oeis.org

0, 0, 1, 2, 5, 7, 13, 18, 27, 38, 54, 71, 99, 131, 172, 226, 295, 379, 488, 621, 788, 998, 1253, 1567, 1955, 2432, 3006, 3712, 4563, 5596, 6840, 8343, 10139, 12306, 14879, 17968, 21635, 26011, 31181, 37330, 44581, 53166, 63259, 75169, 89128, 105554, 124752

Offset: 1

Omar E. Pol, Sep 17 2008

a(n) is also the number of partitions of n with at least one distinct part (i.e., not all parts are equal).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Omar E. Pol, The shell model of partitions

Cf. A000005, A000041, A135010, A138121, A195364.

A182114(n,n-1) = a(n). - Alois P. Heinz, Nov 02 2012

with(numtheory): b:= proc(n) option remember; `if`(n=0, 1, add(add(d, d=divisors(j)) *b(n-j), j=1..n)/n) end: a:= n-> b(n)- tau(n):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 07 2008

Table[PartitionsP[n]-DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Apr 10 2014 *)

al(n)=vector(n,k,numbpart(k)-numdiv(k))

from sympy import npartitions, divisor_count
def A144300(n): return npartitions(n)-divisor_count(n) # Chai Wah Wu, Oct 16 2023

a(n) = p(n) - d(n) = A000041(n) - A000005(n).

A182709 Sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 11, 14, 33, 45, 81, 109, 185, 237, 372, 490, 715, 928, 1326, 1693, 2348, 2998, 4032, 5119, 6795, 8530, 11132, 13952, 17927, 22314, 28417, 35126, 44279, 54532, 68062, 83422, 103427, 126063, 155207, 188506, 230547, 278788, 339223, 408482

Offset: 1

Omar E. Pol, Nov 28 2010, Nov 29 2010

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. For more information see A182699.

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

			For n=7 the partitions of 7 that do not contain "1" as a part are
7
4 + 3
5 + 2
3 + 2 + 2
Then remove one copy of the smallest part of every partition. The rest are the emergent parts:
.,
4, .
5, .
3, 2, .
The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14.
For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.

Row sums of A183152.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
c:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then k
    elif i<2 then 0
    else c(n, i-1, k) +c(n-i, i, i)
      fi
    end:
a:= n-> n*b(n, n) - c(n, n, 0):
seq(a(n), n=1..40);  #  Alois P. Heinz, Dec 01 2010

f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* Harvey P. Dale, Dec 28 2010 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

a(n) = A138880(n) - A182708(n).
a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - Omar E. Pol, Aug 01 2013
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 05 2019

More terms from Alois P. Heinz, Dec 01 2010

A182714 Number of 4's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438

Offset: 1

Omar E. Pol, Nov 13 2011

nonn

Zero together with the first differences of A024788.
Also number of 4's in all partitions of n that do not contain 1 as a part.
a(n) is the number of partitions of n such that m(1) < m(3), where m = multiplicity; e.g., a(7) counts these 3 partitions: [4, 3], [3, 3, 1], [3, 2, 2]. - Clark Kimberling, Apr 01 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 25 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 8                 4's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 2
5 + 3 .......................... 0
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 1
2 + 2 + 2 + 2 .................. 0
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                   1 .......... 0
.                       1 ...... 0
.                           1 .. 0
------------------------------------
.           6 - 3 =              3
.
In the last section of the set of partitions of 8 the difference between the sum of the fourth column and the sum of the fifth column is 6 - 3 = 3 equaling the number of 4's, so a(8) = 3 (see also A024788).
(End)

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

Column 4 of A194812.

Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713, A240058.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..70);  # Alois P. Heinz, Mar 19 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}] (* Clark Kimberling, Apr 01 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i==4, h[[1]], 0]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 4], {n, 52}] (* Robert Price, May 15 2020 *)

A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012

A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208

Offset: 0

Omar E. Pol, Jan 03 2013

nonn

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

			Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _   _ _|_ _|_  |
.                             _ _ _    |  _ _ _    | |
.                   _ _ _ _   _ _ _|_  |  _ _ _|_  | |
.                   _ _    |  _ _    | |  _ _    | | |
.           _ _ _   _ _|_  |  _ _|_  | |  _ _|_  | | |
.     _ _   _ _  |  _ _  | |  _ _  | | |  _ _  | | | |
. _   _  |  _  | |  _  | | |  _  | | | |  _  | | | | |
.  |   | |   | | |   | | | |   | | | | |   | | | | | |
.
. 2    6     12        24         40          70
.
Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
.
11...........................................................
.                                                           /\
.                                                          /  \
.                                                         /    \
7..................................                      /      \
.                                 /\                    /        \
5....................            /  \                /\/          \
.                   /\          /    \          /\  /              \
3..........        /  \        /      \        /  \/                \
2.....    /\      /    \    /\/        \      /                      \
1..  /\  /  \  /\/      \  /            \  /\/                        \
0 /\/  \/    \/          \/              \/                            \
. 0,2,  6,   12,         24,             40,                          70...
.

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Toufik Mansour, Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139582 is twice A000041. A220909 is twice A066186.

Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.

Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016

Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)

a(n) = 2*A006128(n).
a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016

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A182709 Sum of the emergent parts of the partitions of n.

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