A000016 -id:A000016 - cp's OEIS frontend

A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 4, 8, 10, 8, 15, 18, 17, 26, 27, 31, 43, 51, 53, 59, 81, 87, 109, 127, 115, 169, 194, 213, 255, 243, 322, 379, 431, 478, 487, 629, 667, 804, 907, 902, 1151, 1294, 1439, 1530, 1674, 2031, 2290, 2559, 2829, 2973, 3296, 3939

Offset: 0

Gus Wiseman, Jan 21 2023

nonn

			The a(7) = 1 through a(16) = 15 partitions (A=10, B=11, C=12, D=13):
  (421)  (431)  (621)  (532)  (542)  (651)  (643)  (653)  (762)  (754)
         (521)         (541)  (632)  (732)  (652)  (743)  (843)  (763)
                       (631)  (641)  (831)  (742)  (752)  (861)  (853)
                       (721)  (731)  (921)  (751)  (761)  (942)  (862)
                              (821)         (832)  (842)  (A32)  (871)
                                            (841)  (851)  (A41)  (943)
                                            (931)  (932)  (B31)  (952)
                                            (A21)  (941)  (C21)  (961)
                                                   (A31)         (A42)
                                                   (B21)         (A51)
                                                                 (B32)
                                                                 (B41)
                                                                 (C31)
                                                                 (D21)
                                                                 (64321)

This is the strict case of A359896, complement A359895, ranked by A359892.
This is the odd-length case of A359898, complement A359897.
The complement is counted by A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A359893/A359901/A359902 count partitions by median, ranked by A360005.
Cf. A000016, A065795, A066571, A102627, A240850, A240851, A327475, A359894, A359906, A359907, A359910.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]

A054599 a(n) = Sum_{d|n} d*2^(n/d - 1).

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 52, 71, 160, 277, 564, 1035, 2176, 4109, 8348, 16467, 33088, 65553, 131740, 262163, 525456, 1048817, 2099244, 4194327, 8393344, 16777321, 33562676, 67109695, 134234480, 268435485, 536905572, 1073741855, 2147549824

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

			G.f. = x + 4*x^2 + 7*x^3 + 16*x^4 + 21*x^5 + 52*x^6 + 71*x^7 + 160*x^8 + 277*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..3300
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A000016, A000031, A054598, A054600, A054601.

{0}~Join~Table[DivisorSum[n, 2^(n/# - 1) # &], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 20 2015 *)
Table[SeriesCoefficient[-Log[-QPochhammer[2, x]] n/2, {x, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = if (n<1, 0, sumdiv(n, d, d*2^(n/d - 1))); \\ Michel Marcus, Nov 21 2015

G.f.: Sum_{n>0} n*x^n/(1-2*x^n). - Vladeta Jovovic, Oct 27 2002
L.g.f.: -log(-(2;-x)inf)/2, where (a;q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2015
G.f.: Sum_{k>=1} 2^(k-1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 10 2019
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 16 2019

A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).

Original entry on oeis.org

1, 0, 2, 4, 4, 0, 10, 32, 30, 0, 94, 344, 316, 0, 1096, 4096, 3856, 0, 13798, 52432, 49940, 0, 182362, 699072, 671092, 0, 2485534, 9586984, 9256396, 0, 34636834, 134217728, 130150588, 0, 490853416, 1908874584, 1857283156, 0, 7048151672, 27487790720

Offset: 1

Seiichi Manyama, Feb 28 2018

nonn

Apparently a(2*n + 1) = A053656(2*n + 1) for n >= 0. - Georg Fischer, Mar 26 2019

			Solutions for n = 7:
--------------------------
1 +2 +3 +4 +5 +6 +7 =  28.
1 +2 +3 +4 +5 +6 -7 =  14.
1 +2 -3 +4 -5 -6 +7 =   0.
1 +2 -3 +4 -5 -6 -7 = -14.
1 +2 -3 -4 +5 +6 +7 =  14.
1 +2 -3 -4 +5 +6 -7 =   0.
1 -2 +3 +4 -5 +6 +7 =  14.
1 -2 +3 +4 -5 +6 -7 =   0.
1 -2 -3 -4 -5 +6 +7 =   0.
1 -2 -3 -4 -5 +6 -7 = -14.

Seiichi Manyama, Table of n, a(n) for n = 1..3334 (terms 1..1000 from Alois P. Heinz)

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): this sequence (k=1), A300268 (k=2), A300269 (k=3).
Cf. A000016, A026119, A053656, A058377, A063776, A300307, A300328.
Cf. A016825 (4n+2).

b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),
      add(b(irem(n+j, m), i-1, m), j=[i, m-i]))
    end:
a:= n-> b(0, n-1, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 01 2018

b[n_, i_, m_] := b[n, i, m] = If[i == 0, If[n == 0, 1, 0], Sum[b[Mod[n + j, m], i - 1, m], {j, {i, m - i}}]];
a[n_] := b[0, n - 1, n];
Array[a, 60] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    i1 = 2 * i
    a = ary.clone
    (0..n - 1).each{|j| a[(j + i1) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) / 2) % n] / 2
end
def A300190(n)
  (1..n).map{|i| A(i)}
end
p A300190(100)

a(4*n+1) = A000016(n), a(4*n+2) = 0, a(4*n+3) = A000016(n), a(4*n+4) = 2 * A000016(n) for n > 0.

a(2^n) = 2^A000325(n) for n > 1.

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

A053633 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1+x^j) mod x^(n+1)-1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 3, 3, 3, 3, 6, 5, 5, 6, 5, 5, 10, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 30, 28, 28, 29, 28, 28, 29, 28, 28, 52, 51, 51, 51, 51, 52, 51, 51, 51, 51, 94, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 172, 170, 170, 172, 170, 170, 172

Offset: 0

N. J. A. Sloane, Mar 22 2000

T(n,k) = number of binary vectors (x_1,...,x_n) satisfying Sum_{i=1..n} i*x_i = k (mod n+1) = size of Varshamov-Tenengolts code VT_k(n).

			Triangle begins:
  k  0    1    2    3    4    5    6    7    8    9
n
0    1;
1    1,   1;
2    2,   1,   1;
3    2,   2,   2,   2;
4    4,   3,   3,   3,   3;
5    6,   5,   5,   6,   5,   5;
6   10,   9,   9,   9,   9,   9,   9;
7   16,  16,  16,  16,  16,  16,  16,  16;
8   30,  28,  28,  29,  28,  28,  29,  28,  28;
9   52,  51,  51,  51,  51,  52,  51,  51,  51,  51;
    ...
[Edited by _Seiichi Manyama_, Mar 11 2018]

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.

Seiichi Manyama, Rows n = 0..139, flattened
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
N. J. A. Sloane, On single-deletion-correcting codes
Index entries for sequences related to subset sums modulo m
Index entries for sequences related to Gijswijt's sequence

Cf. A053632, A063776, A300328, A300628. Leading coefficients give A000016, next column gives A000048.

with(numtheory): A053633 := proc(n,k) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+(1/(2*n))*2^(n/d)*phi(d)*mobius(d/gcd(d,k))/phi(d/gcd(d,k)); fi; od; t1; end;

Flatten[ Table[ CoefficientList[ PolynomialMod[ Product[1+x^j, {j,1,n}], x^(n+1)-1], x], {n,0,11}]] (* Jean-François Alcover, May 04 2011 *)

The Maple code gives an explicit formula.

A053634 a(n) = Sum_{ d divides n } phi(d)*2^(n/d)/(2n).

Original entry on oeis.org

2, 3, 4, 7, 10, 18, 30, 54, 94, 176, 316, 591, 1096, 2058, 3856, 7301, 13798, 26244, 49940, 95373, 182362, 349626, 671092, 1290714, 2485534, 4793790, 9256396, 17896284, 34636834, 67109898, 130150588, 252647064, 490853416, 954440950

Offset: 3

N. J. A. Sloane, Mar 23 2000

nonn

Offset is 3 because a(2) is 3/2, not an integer. - Michel Marcus, Sep 11 2013

G. C. Greubel, Table of n, a(n) for n = 3..3000
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A000016, A000031, A053635, A063776.

a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2n); Table[a[n], {n, 3, 36}] (* Jean-François Alcover, Dec 07 2015 *)

a(n) = sumdiv (n, d, eulerphi(d)*2^(n/d)/(2*n));  \\ Michel Marcus, Sep 11 2013

a(n) = A000031(n)/2.

A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).

Original entry on oeis.org

0, 2, 8, 14, 32, 42, 104, 142, 320, 554, 1128, 2070, 4352, 8218, 16696, 32934, 66176, 131106, 263480, 524326, 1050912, 2097634, 4198488, 8388654, 16786688, 33554642, 67125352, 134219390, 268468960, 536870970, 1073811144, 2147483710, 4295099648, 8589940890

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

Row sums of A322200, where A322200 describes Sum_{n>=1} -log(1 - (x^n + y^n)). - Paul D. Hanna, Dec 01 2018

Cf. A054599, A054600, A054601, A000016, A000031.

Cf. A322200.

Table[CoefficientList[Series[-Log[-QPochhammer[2, x]], {x, 0, 60}], x][[n]] (n - 1), {n, 1, 60}] (* Benedict W. J. Irwin, Jun 23 2016 *)

a(n) = sumdiv(n, d, d*2^(n/d)); \\ Michel Marcus, Jul 01 2016

L.g.f.: -log(Product_{ k>0 } (1-2*x^k)) = Sum_{ n>=0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jun 23 2016

G.f.: Sum_{k>=1} 2^k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018

A327471 Number of subsets of {1..n} not containing their mean.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 48, 102, 214, 440, 900, 1830, 3706, 7486, 15092, 30380, 61100, 122780, 246566, 494912, 992984, 1991620, 3993446, 8005388, 16044460, 32150584, 64414460, 129037790, 258462026, 517641086, 1036616262, 2075721252, 4156096036, 8320912744, 16658202200

Offset: 0

Gus Wiseman, Sep 12 2019

nonn

			The a(1) = 1 through a(5) = 22 subsets:
  {}  {}     {}     {}         {}
      {1,2}  {1,2}  {1,2}      {1,2}
             {1,3}  {1,3}      {1,3}
             {2,3}  {1,4}      {1,4}
                    {2,3}      {1,5}
                    {2,4}      {2,3}
                    {3,4}      {2,4}
                    {1,2,4}    {2,5}
                    {1,3,4}    {3,4}
                    {1,2,3,4}  {3,5}
                               {4,5}
                               {1,2,4}
                               {1,2,5}
                               {1,3,4}
                               {1,4,5}
                               {2,3,5}
                               {2,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}

Subsets containing their mean are A065795.
Subsets containing n but not their mean are A327477.
Partitions not containing their mean are A327472.
Strict partitions not containing their mean are A240851.
Cf. A000016, A007865, A051293, A067538, A082550, A114639, A135342, A324741, A327476.

Table[Length[Select[Subsets[Range[n]],!MemberQ[#,Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327471(n): return (1<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1) # Chai Wah Wu, Feb 22 2023

a(n) = 2^n - A065795(n). - Alois P. Heinz, Sep 13 2019

More terms from Alois P. Heinz, Sep 13 2019

A327474 Number of distinct means of subsets of {1..n}, where {} has mean 0.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 38, 56, 78, 106, 138, 180, 226, 284, 348, 420, 500, 596, 698, 818, 946, 1086, 1236, 1408, 1588, 1788, 2000, 2230, 2472, 2742, 3020, 3328, 3652, 3996, 4356, 4740, 5136, 5568, 6018, 6492, 6982, 7512, 8054, 8638, 9242, 9870, 10520, 11216

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(3) = 6 distinct means are 0, 1, 3/2, 2, 5/2, 3.

The version for only nonempty subsets is A135342.

Cf. A000016, A051293, A065795, A082550, A327475, A327481.

a:= proc(n) option remember; `if`(n<4, [1, 2, 4, 6][n+1],
      2*a(n-1)-a(n-2)+numtheory[phi](n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 22 2023

Table[Length[Union[Mean/@Subsets[Range[n]]]],{n,0,10}]

from itertools import count, islice
from sympy import totient
def A327474_gen(): # generator of terms
    a, b = 4, 6
    yield from (1,2,4,6)
    for n in count(3):
        a, b = b, (b<<1)-a+totient(n)
        yield b
A327474_list = list(islice(A327474_gen(),30)) # Chai Wah Wu, Feb 22 2023

a(n) = A135342(n) + 1.

a(n) = 2*a(n-1) - a(n-2) + phi(n-1) for n>3. - Chai Wah Wu, Feb 22 2023

A363526 Number of integer partitions of n with reverse-weighted sum 3*n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 2, 4, 4, 4, 5, 5, 4, 7, 7, 5, 8, 7, 6, 11, 9, 8, 11, 10, 10, 13, 12, 11, 15, 15, 12, 17, 16, 14, 20, 18, 16, 22, 20, 19, 24, 22, 20, 27, 26, 23, 29, 27, 25, 33, 30, 28, 35, 33, 31, 38, 36, 33, 41, 40

Offset: 0

Gus Wiseman, Jun 10 2023

nonn

Are the partitions counted all of length 4 or 5?

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

			The partition (6,4,4,1) has sum 15 and reverse-weighted sum 45 so is counted under a(15).
The a(n) partitions for n = {5, 10, 15, 16, 21, 24}:
  (1,1,1,1,1)  (4,3,2,1)    (6,4,4,1)    (6,5,4,1)  (8,6,6,1)   (9,7,7,1)
               (2,2,2,2,2)  (6,5,2,2)    (6,6,2,2)  (8,7,4,2)   (9,8,5,2)
                            (7,3,3,2)    (7,4,3,2)  (9,5,5,2)   (9,9,3,3)
                            (3,3,3,3,3)             (9,6,3,3)   (10,6,6,2)
                                                    (10,4,4,3)  (10,7,4,3)
                                                                (11,5,5,3)
                                                                (12,4,4,4)

Positions of terms with omega > 4 appear to be A079998.
The version for compositions is A231429.
The non-reverse version is A363527.
These partitions have ranks A363530, reverse A363531.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528.

Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==3n&]],{n,0,30}]

cp's OEIS Frontend

A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

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A054599 a(n) = Sum_{d|n} d*2^(n/d - 1).

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A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).

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A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

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A053633 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1+x^j) mod x^(n+1)-1.

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A053634 a(n) = Sum_{ d divides n } phi(d)*2^(n/d)/(2n).

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A054598 a(0)=0; for n>0, a(n) = Sum_{d|n} d*2^(n/d).

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A327471 Number of subsets of {1..n} not containing their mean.

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