A141285 -id:A141285 - cp's OEIS frontend

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

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

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

A194602 Integer partitions interpreted as binary numbers.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 21, 23, 27, 31, 43, 47, 55, 63, 85, 87, 91, 95, 111, 119, 127, 171, 175, 183, 191, 219, 223, 239, 255, 341, 343, 347, 351, 367, 375, 383, 439, 447, 479, 495, 511, 683, 687, 695, 703, 731, 735, 751, 767, 879, 887, 895, 959, 991, 1023, 1365, 1367, 1371, 1375, 1391

Offset: 0

Tilman Piesk, Aug 30 2011

The 2^(n-1) compositions of n correspond to binary numbers, and the partitions of n can be seen as compositions with addends ordered by size, so they also correspond to binary numbers.
The finite sequence for partitions of n (ordered by size) is the beginning of the sequence for partitions of n+1, which leads to an infinite sequence.
From Tilman Piesk, Jan 30 2016: (Start)
It makes sense to regard the positive values as a triangle with row lengths A002865(n) and row numbers n>=2. In this triangle row n contains all partitions of n with non-one addends only. See link "Triangle with Young diagrams".
This sequence contains all binary palindromes with m runs of n ones separated by single zeros. They are ordered in the array A249544. All the rows and columns of this array are subsequences of this sequence, notably its top row (A000225, the powers of two minus one).
Sequences by Omar E. Pol: The "triangle" A210941 defines the same sequence of partitions. Its n-th row shows the non-one addends of the n-th partition. There are A194548(n) of them, and A141285(n) is the largest among them. (The "triangle" A210941 does not actually form a triangle, but A210941 and A141285 do.) Note that the offset of these sequences is 1 and not 0.
(End)
Numbers whose binary representation has runs of '1's of weakly increasing length (with trailing '0's (introducing a run of length 0) forbidden, i.e., only odd terms beyond 0). - M. F. Hasler, May 14 2020

			From _Joerg Arndt_, Nov 17 2012: (Start)
With leading zeros included, the first A000041(n) terms correspond to the list of partitions of n as nondecreasing compositions in lexicographic order.
For example, the first A000041(10)=42 terms correspond to the partitions of 10 as follows (dots for zeros in the binary expansions):
[ n]   binary(a(n))  a(n)  partition
[ 0]   ..........     0    [ 1 1 1 1 1 1 1 1 1 1 ]
[ 1]   .........1     1    [ 1 1 1 1 1 1 1 1 2 ]
[ 2]   ........11     3    [ 1 1 1 1 1 1 1 3 ]
[ 3]   .......1.1     5    [ 1 1 1 1 1 1 2 2 ]
[ 4]   .......111     7    [ 1 1 1 1 1 1 4 ]
[ 5]   ......1.11    11    [ 1 1 1 1 1 2 3 ]
[ 6]   ......1111    15    [ 1 1 1 1 1 5 ]
[ 7]   .....1.1.1    21    [ 1 1 1 1 2 2 2 ]
[ 8]   .....1.111    23    [ 1 1 1 1 2 4 ]
[ 9]   .....11.11    27    [ 1 1 1 1 3 3 ]
[10]   .....11111    31    [ 1 1 1 1 6 ]
[11]   ....1.1.11    43    [ 1 1 1 2 2 3 ]
[12]   ....1.1111    47    [ 1 1 1 2 5 ]
[13]   ....11.111    55    [ 1 1 1 3 4 ]
[14]   ....111111    63    [ 1 1 1 7 ]
[15]   ...1.1.1.1    85    [ 1 1 2 2 2 2 ]
[16]   ...1.1.111    87    [ 1 1 2 2 4 ]
[17]   ...1.11.11    91    [ 1 1 2 3 3 ]
[18]   ...1.11111    95    [ 1 1 2 6 ]
[19]   ...11.1111   111    [ 1 1 3 5 ]
[20]   ...111.111   119    [ 1 1 4 4 ]
[21]   ...1111111   127    [ 1 1 8 ]
[22]   ..1.1.1.11   171    [ 1 2 2 2 3 ]
[23]   ..1.1.1111   175    [ 1 2 2 5 ]
[24]   ..1.11.111   183    [ 1 2 3 4 ]
[25]   ..1.111111   191    [ 1 2 7 ]
[26]   ..11.11.11   219    [ 1 3 3 3 ]
[27]   ..11.11111   223    [ 1 3 6 ]
[28]   ..111.1111   239    [ 1 4 5 ]
[29]   ..11111111   255    [ 1 9 ]
[30]   .1.1.1.1.1   341    [ 2 2 2 2 2 ]
[31]   .1.1.1.111   343    [ 2 2 2 4 ]
[32]   .1.1.11.11   347    [ 2 2 3 3 ]
[33]   .1.1.11111   351    [ 2 2 6 ]
[34]   .1.11.1111   367    [ 2 3 5 ]
[35]   .1.111.111   375    [ 2 4 4 ]
[36]   .1.1111111   383    [ 2 8 ]
[37]   .11.11.111   439    [ 3 3 4 ]
[38]   .11.111111   447    [ 3 7 ]
[39]   .111.11111   479    [ 4 6 ]
[40]   .1111.1111   495    [ 5 5 ]
[41]   .111111111   511    [ 10 ]
(End)

Tilman Piesk, Table of n, a(n) for n = 0..8348
Tilman Piesk, Same table with binary strings and non-one addends
Tilman Piesk, Triangle with Young diagrams (n = 2..20).
Tilman Piesk, Integer partitions and Permutations and partitions in the OEIS
Tilman Piesk, Python functions, keynum_to_valnum(n) = a(n), valnum_to_keynum(a(n)) = n.
Li-yao Xia, Identities for A194602
Index entries for sequences related to binary expansion of n

Cf. A000041 (partition numbers).
Cf. A002865 (row lengths).
Cf. A002450, A000225 (subsequences).
Cf. A249544 (rows and columns are subsequences).
Cf. A210941, A194548, A141285, A228354, A326956.

lim = 12;
Sort[FromDigits[Reverse@ #, 2] & /@
   Map[If[Length@ # == 0, {0}, Flatten@ Most@ #] &@
     Riffle[#, Table[0, Length@ #]] &,
     Map[Table[1, # - 1] &,
       Sort@ IntegerPartitions@ lim /. 1 -> Nothing, {2}]]]
(* Michael De Vlieger, Feb 14 2016 *)

isA194602(n) = if(!n,1,if(!(n%2),0,my(prl=0,rl=0); while(n, if(0==(n%2),if((prl && rl>prl)||0==(n%4), return(0)); prl=rl; rl=0, rl++); n >>= 1); ((0==prl)||(rl<=prl)))); \\ - Antti Karttunen, Dec 06 2021

a( A000041(n)-1 ) = A000225(n-1) for n>=1. - Tilman Piesk, Apr 16 2012
a( A000041(2n-1) ) = A002450(n)  for n>=1. - Tilman Piesk, Apr 16 2012
a( A249543 ) = A249544. - Tilman Piesk, Oct 31 2014
a(n) = A228354(1+n) - 1. - Antti Karttunen, Dec 06 2021

Comments edited by Li-yao Xia, May 13 2014

Incorrect PARI-program removed by Antti Karttunen, Dec 09 2021

A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 14, 15, 19, 24, 27, 28, 33, 40, 42, 43, 47, 49, 52, 53, 59, 70, 73, 74, 79, 81, 85, 86, 93, 108, 110, 111, 115, 117, 120, 121, 127, 131, 136, 137, 141, 142, 150, 172, 175, 176, 181, 183, 187, 188, 195, 199, 202, 203, 209, 211, 216, 217, 226, 256

Offset: 0

Omar E. Pol, Jul 28 2013

nonn

This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.
To define the sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks.
If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).
If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).
The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.
Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) 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, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.

			For n = 30 the structure has 108 toothpicks, so a(30) = 108.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15  _ _ _ _      |
4 + 3                               _ _ _ _|_    |
5 + 2                               _ _ _    |   |
3 + 2 + 2                           _ _ _|_ _|_  |
6 + 1                           11  _ _ _      | |
3 + 3 + 1                           _ _ _|_    | |
4 + 2 + 1                           _ _    |   | |
2 + 2 + 2 + 1                       _ _|_ _|_  | |
5 + 1 + 1                        7  _ _ _    | | |
3 + 2 + 1 + 1                       _ _ _|_  | | |
4 + 1 + 1 + 1                    5  _ _    | | | |
2 + 2 + 1 + 1 + 1                   _ _|_  | | | |
3 + 1 + 1 + 1 + 1                3  _ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2  _  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1   | | | | | | |
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms:
.
.                              _ _ _    _ _ _
.                _ _   _ _     _ _      _ _  |
.      _    _    _     _  |    _  |     _  | |
.            |    |     | |     | |      | | |
.
.      1    2     4     6       9        12
.
.
.                          _ _ _ _     _ _ _ _
.      _ _       _ _       _ _         _ _    |
.      _ _ _     _ _|_     _ _|_       _ _|_  |
.      _ _  |    _ _  |    _ _  |      _ _  | |
.      _  | |    _  | |    _  | |      _  | | |
.       | | |     | | |     | | |       | | | |
.
.        14        15         19          24
.
.
.                          _ _ _ _ _    _ _ _ _ _
.    _ _ _      _ _ _      _ _ _        _ _ _    |
.    _ _ _ _    _ _ _|_    _ _ _|_      _ _ _|_  |
.    _ _    |   _ _    |   _ _    |     _ _    | |
.    _ _|_  |   _ _|_  |   _ _|_  |     _ _|_  | |
.    _ _  | |   _ _  | |   _ _  | |     _ _  | | |
.    _  | | |   _  | | |   _  | | |     _  | | | |
.     | | | |    | | | |    | | | |      | | | | |
.
.       27         28         33            40
.
Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
.
7                                    33
.                                    /\
5                      19           /  \
.                      /\          /    \
3            9        /  \     27 /      \
2       4    /\   14 /    \    /\/        \
1    1  /\  /  \  /\/      \  / 28         \
.    /\/  \/    \/ 15       \/              \
.   0  2   6    12          24              40
.

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. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.

a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1.

A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

Original entry on oeis.org

1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470

Offset: 0

Omar E. Pol, Jul 29 2013

nonn

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

			For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15 |_ _ _ _      |
4 + 3                              |_ _ _ _|_    |
5 + 2                              |_ _ _    |   |
3 + 2 + 2                          |_ _ _|_ _|_  |
6 + 1                           11 |_ _ _      | |
3 + 3 + 1                          |_ _ _|_    | |
4 + 2 + 1                          |_ _    |   | |
2 + 2 + 2 + 1                      |_ _|_ _|_  | |
5 + 1 + 1                        7 |_ _ _    | | |
3 + 2 + 1 + 1                      |_ _ _|_  | | |
4 + 1 + 1 + 1                    5 |_ _    | | | |
2 + 2 + 1 + 1 + 1                  |_ _|_  | | | |
3 + 1 + 1 + 1 + 1                3 |_ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_|
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                        |_ _ _      |
.                                        |_ _ _|_    |
.                                        |_ _    |   |
.                             _ _ _ _ _  |_ _|_ _|_  |
.                            |_ _ _    | |_ _ _    | |
.                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | |
.                  |_ _    | |_ _    | | |_ _    | | |
.           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | |
.     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | |
. _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4    10     18       33         52          87

Omar E. Pol, Illustration of the seven regions of 5
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

Original entry on oeis.org

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

Vladeta Jovovic, Oct 07 2003

nonn

Essentially first differences of A024786 (see the formula). Also, a(n) is the number of 2's in the last section of the set of partitions of n+2 (see A135010). - Omar E. Pol, Sep 10 2008
From Gus Wiseman, May 20 2024: (Start)
Also the number of integer partitions of n containing an even number of ones, ranked by A003159. The a(0) = 1 through a(8) = 15 partitions are:
  ()  .  (2)   (3)  (4)     (5)    (6)       (7)      (8)
         (11)       (22)    (32)   (33)      (43)     (44)
                    (211)   (311)  (42)      (52)     (53)
                    (1111)         (222)     (322)    (62)
                                   (411)     (511)    (332)
                                   (2211)    (3211)   (422)
                                   (21111)   (31111)  (611)
                                   (111111)           (2222)
                                                      (3311)
                                                      (4211)
                                                      (22211)
                                                      (41111)
                                                      (221111)
                                                      (2111111)
                                                      (11111111)
Also the number of integer partitions of n + 1 containing an odd number of ones, ranked by A036554.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A024786, A135010, A138121, A141285.
The unsigned version is A000070, strict A036469.
For powers of 2 instead number of partitions we have A001045.
The strict or odd version is A025147 or A096765.
The ordered version (compositions instead of partitions) is A078008.
For powers of 2 instead of -1 we have A259401, cf. A259400.
A002865 counts partitions with no ones, column k=0 of A116598.
A072233 counts partitions by sum and length.
Cf. A000009, A003159, A026804, A027187, A027193, A036554, A058695, A101707.

Table[Sum[(-1)^(n-k)*PartitionsP[k], {k,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 16 2015 *)
(* more efficient program *) sig = 1; su = 1; Flatten[{1, Table[sig = -sig; su = su + sig*PartitionsP[n]; Abs[su], {n, 1, 50}]}] (* Vaclav Kotesovec, Nov 06 2016 *)
Table[Length[Select[IntegerPartitions[n], EvenQ[Count[#,1]]&]],{n,0,30}] (* Gus Wiseman, May 20 2024 *)

from sympy import npartitions
def A087787(n): return sum(-npartitions(k) if n-k&1 else npartitions(k) for k in range(n+1)) # Chai Wah Wu, Oct 25 2023

G.f.: 1/(1+x)*1/Product_{k>0} (1-x^k).
a(n) = 1/n*Sum_{k=1..n} (sigma(k)+(-1)^k)*a(n-k).
a(n) = A024786(n+2)-A024786(n+1). - Omar E. Pol, Sep 10 2008
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) - (11/16 + (23*Pi^2)/6912)/n). - Vaclav Kotesovec, Nov 05 2016
a(n) = A000041(n) - a(n-1). - Jon Maiga, Aug 29 2019
Alternating partial sums of A000041. - Gus Wiseman, May 20 2024

A182982 Triangle read by rows: row n lists the parts of the n-th shell of the table A182742.

Original entry on oeis.org

2, 2, 4, 2, 2, 3, 3, 6, 2, 2, 2, 2, 3, 5, 4, 4, 8, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 7, 4, 6, 5, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 6, 3, 4, 5, 3, 9, 4, 4, 4, 4, 8, 5, 7, 6, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Jan 26 2011

Apparently this is the main table for even numbers of the shell model of partitions. It appears that the table shows an overlapping of all the heads of last sections of partitions of all even numbers. This is the table 2.0 mentioned in A135010, a geometric version of the table A182742. For odd numbers see A182983. The largest parts of the rows of the diagram give A182732.

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

Cf. A135010, A141285, A182732, A182742, A182980, A182983.

A182983 Triangle read by rows: row n lists the parts of the n-th shell of the table A182743.

Original entry on oeis.org

3, 2, 5, 2, 2, 3, 4, 7, 2, 2, 2, 2, 3, 3, 3, 3, 6, 4, 5, 9, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 3, 4, 4, 3, 8, 4, 7, 5, 6, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 3, 3, 7, 3, 4, 6, 3, 5, 5, 3, 10, 4, 4, 5, 4, 9, 5, 8, 6, 7, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Jan 26 2011

Apparently this is the main table for odd numbers of the shell model of partitions. It appears that the table shows an overlapping of all the heads of last sections of partitions of all odd numbers. This is the table 2.1 mentioned in A135010, a geometric version of the table A182743. For even numbers see A182982. The largest parts of the rows of the diagram give A182733.

			Triangle begins:
3,
2, 5,
2, 2, 3, 4, 7,
2, 2, 2, 2, 3, 3, 3, 3, 6, 4, 5, 9

Cf. A135010, A141285, A182733, A182743, A182980, A182982.

A207779 Largest part plus the number of parts of the n-th region of the section model of partitions.

Original entry on oeis.org

2, 4, 6, 3, 9, 4, 12, 3, 6, 4, 17, 4, 7, 5, 22, 3, 6, 4, 10, 6, 5, 30, 4, 7, 5, 11, 4, 8, 6, 39, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52, 4, 7, 5, 11, 4, 8, 6, 17, 6, 5, 11, 8, 7, 67, 3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 22, 4, 8, 6, 13, 5, 10, 8

Offset: 1

Omar E. Pol, Mar 08 2012

Also semiperimeter of the n-th region of the geometric version of the section model of partitions. Note that a(n) is easily viewable as the sum of two perpendicular segments with a shared vertex. The horizontal segment has length A141285(n) and the vertical segment has length A194446(n). The difference between these two segments gives A194447(n). See also an illustration in the Links section. For the definition of "region" see A206437.

Also triangle read by rows: T(n,k) = largest part plus the number of parts of the k-th region of the last section of the set of partitions of n.

			Written as a triangle begins:
2;
4;
6;
3, 9;
4, 12;
3, 6, 4, 17;
4, 7, 5, 22;
3, 6, 4, 10, 6, 5, 30;
4, 7, 5, 11, 4, 8, 6, 39;
3, 6, 4, 10, 6, 5, 15, 5, 9, 7, 6, 52;

Omar E. Pol, Illustration of the seven regions of 5

Row n has length A187219(n). Last term of row n is A133041(n). Where record occur give A000041, n >= 1.

Cf. A002865, A135010, A182699, A182709, A183152, A194436, A194437, A194438, A194439, A194447, A206437.

a(n) = A141285(n) + A194446(n).

A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 0, 1, 7, 1, 2, 0, 1, 9, 6, 2, 2, 0, 1, 15, 4, 4, 1, 2, 0, 1, 19, 13, 4, 5, 1, 2, 0, 1, 32, 10, 10, 3, 4, 1, 2, 0, 1, 40, 24, 10, 9, 4, 4, 1, 2, 0, 1, 60, 23, 18, 8, 8, 3, 4, 1, 2, 0, 1, 78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1

Offset: 1

Omar E. Pol, Feb 17 2012

For the calculation of row n, the number of odd/even parts, etc, take the row n from the triangle A207031 and then follow the same rules of A206563.

			Triangle begins:
  1;
  1,   1;
  3,   0,  1;
  3,   3,  0,  1;
  7,   1,  2,  0, 1;
  9,   6,  2,  2, 0, 1;
  15,  4,  4,  1, 2, 0, 1;
  19, 13,  4,  5, 1, 2, 0, 1;
  32, 10, 10,  3, 4, 1, 2, 0, 1;
  40, 24, 10,  9, 4, 4, 1, 2, 0, 1;
  60, 23, 18,  8, 8, 3, 4, 1, 2, 0, 1;
  78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1;

Cf. A006128, A066897, A066898, A135010, A138121, A138135, A138137, A141285, A181187, A182703, A206563, A207031.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A207031(n,j)).

It appears that A182703(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

cp's OEIS Frontend

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

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

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A194602 Integer partitions interpreted as binary numbers.

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A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

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A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

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A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

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A182982 Triangle read by rows: row n lists the parts of the n-th shell of the table A182742.

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A182983 Triangle read by rows: row n lists the parts of the n-th shell of the table A182743.

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A207779 Largest part plus the number of parts of the n-th region of the section model of partitions.