A116608 -id:A116608 - cp's OEIS frontend

A376311 Position of first appearance of n in the sequence of first differences of squarefree numbers, or the sequence ends if there is none.

1, 3, 6, 31, 150, 515, 13391, 131964, 664313, 5392318, 159468672, 134453711, 28728014494, 50131235121, 634347950217, 48136136076258, 1954623227727573, 14433681032814706, 76465679305346797

Offset: 1

Gus Wiseman, Sep 22 2024

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
The positions of first appearances are a(n).

This is the position of first appearance of n in A076259, ones A375927.
For compression instead of positions of first appearances we have A376305.
For run-lengths instead of first appearances we have A376306.
For run-sums instead of first appearances we have A376307.
For prime-powers instead of squarefree numbers we have A376341.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A120992, A375707, A376312, A376342.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[10000],SquareFreeQ]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

a(11)-a(19) from Amiram Eldar, Sep 24 2024

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 0, 0, -1, -3, -7, -14, -25, -41, -64, -100, -165, -294, -550, -1023, -1795, -2823, -3658, -2882, 2873, 20435, 62185, 148863, 314008, 613957, 1155794, 2175823, 4244026, 8753538, 19006490, 42471787, 95234575, 210395407, 453413866, 949508390, 1931939460

Offset: 0

Vaclav Kotesovec, Oct 09 2017

sign

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph: a(n)/2^n (40000 terms)

Cf. A025147, A266232, A294467, A294468.
Cf. A218481, A294466, A095051.
The non-strict version is the absolute value of A281425; see A175804, A320590.
Up to sign, same as A380412. See A320591, A377285, A378970, A378971.
A000009 counts strict integer partitions, differences A087897.
Cf. A000041, A002865, A007442, A008284, A116608.

Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

A361849 Number of integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 3, 4, 7, 9, 9, 15, 16, 20, 26, 34, 37, 50, 55, 68, 86, 103, 117, 145, 168, 201, 236, 282, 324, 391, 449, 525, 612, 712, 818, 962, 1106, 1278, 1470, 1698, 1939, 2238, 2550, 2924, 3343, 3824, 4341, 4963, 5627, 6399, 7256, 8231, 9300

Offset: 1

Gus Wiseman, Apr 02 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(4) = 1 through a(11) = 9 partitions:
  211  2111  21111  421     422      4221      631        632
                    3211    221111   4311      4222       5321
                    22111   2111111  2211111   42211      5411
                    211111           21111111  322111     42221
                                               2221111    43211
                                               22111111   332111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
For example, the partition (3,2,1,1) has maximum 3 and median 3/2, so is counted under a(7).

For minimum instead of median we have A118096.
For length instead of median we have A237753.
This is the equal case of A361848.
For mean instead of median we have A361853.
These partitions have ranks A361856.
For "greater" instead of "equal" we have A361857, allowing equality A361859.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
A361860 counts partitions with minimum equal to median.
Cf. A013580, A027193, A053263, A061395, A067659, A111907, A116608, A237755, A240219, A361851.

Table[Length[Select[IntegerPartitions[n],Max@@#==2*Median[#]&]],{n,30}]

A056556 First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Henry Bottomley, Jun 26 2000

If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n).
From Gus Wiseman, Jul 03 2019: (Start)
Also the maximum number of distinct multiplicities among integer partitions of n. For example, random partitions of 56 realizing each number of distinct multiplicities are:
  1: (24,17,6,5,3,1)
  2: (10,9,9,5,5,4,4,3,3,2,1,1)
  3: (6,5,5,5,4,4,4,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  4: (28,5,5,3,3,3,2,2,1,1,1,1,1)
  5: (13,4,4,4,4,4,3,3,3,2,2,2,2,2,2,1,1)
  6: (6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1)
The maximum number of distinct multiplicities is 6, so a(56) = 6.
(End)

			3 is (3+1) * (3+2)/2 = 10 times in the sequence all these occurrences are in consecutive places. The first 3 is at position binomial(3 + 2, 3) = 10, the last one at binomial((3 + 1) + 2, 3) - 1. - _David A. Corneth_, Oct 14 2022

Cf. A000292, A003056, A056557, A056558, A056559, A056560.
See also A194847.
Cf. A088880, A088881, A116608, A325242.

Table[Table[m, {(m+1)(m+2)/2}], {m, 0, 7}] // Flatten (* Jean-François Alcover, Feb 28 2019 *)

a(n)=my(t=polrootsreal(x^3+3*x^2+2*x-6*n)); t[#t]\1 \\ Charles R Greathouse IV, Feb 22 2017

from math import comb
from sympy import integer_nthroot
def A056556(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(nChai Wah Wu, Nov 04 2024

a(n) = floor(x) where x is the (largest real) solution to x^3 + 3x^2 + 2x - 6n = 0; a(A000292(n)) = n+1.
a(n+1) = a(n)+1 if a(n) = A056558(n), otherwise a(n). - Graeme McRae, Jan 09 2007
a(n) = floor(t/3 + 1/t - 1), where t = (81*n + 3*sqrt(729*n^2 - 3))^(1/3). - Ridouane Oudra, Mar 21 2021
a(n) = floor(t + 1/(3*t) - 1), where t = (6*n)^(1/3), for n>=1. - Ridouane Oudra, Nov 04 2022
a(n) = m if n>=binomial(m+2,3) and a(n) = m-1 otherwise where m = floor((6n+6)^(1/3)). - Chai Wah Wu, Nov 04 2024

Incorrect formula deleted by Ridouane Oudra, Nov 04 2022

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of partitions is A000726.
The version for run-lengths instead of runs is A032020.
These words are ranked by A175413.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329738 counts compositions with equal run-lengths.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]

P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]

A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 28 2023

I call this a tetrangle because it is a sequence of finite triangles. - Gus Wiseman, Jan 30 2023

			Tetrangle begins:
  1   1     1       1         1           1             1
      1 0   0 1     1 1       0 2         1 2           0 3
            1 0 0   0 1 0     0 2 0       1 1 1         0 3 1
                    1 0 0 0   0 1 0 0     0 2 0 0       0 2 1 0
                              1 0 0 0 0   0 1 0 0 0     0 2 0 0 0
                                          1 0 0 0 0 0   0 1 0 0 0 0
                                                        1 0 0 0 0 0 0
For example, finite triangle n = 5 counts the following partitions:
    (5)
     .    (41)(32)
     .   (311)(221)  .
     .     (2111)    .   .
  (11111)     .      .   .   .

Row sums are A008284 (partitions by number of parts), reverse A058398.
First columns i = 1 are A051731.
Last columns i = k are A060016.
Column sums are A116608 (partitions by number of distinct parts).
Positive terms are counted by A360072.
A000041 counts partitions, strict A000009.
Other tetrangles: A318393, A318816, A320808, A334433, A345197.
Cf. A055884, A327482, A331195, A360010, A360069.

Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]],{n,1,9},{k,1,n},{i,1,k}]

A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 10, 13, 20, 26, 39, 50, 71, 87, 121, 156, 208, 265, 348, 440, 566, 712, 906, 1131, 1424, 1766, 2224, 2738, 3390, 4168, 5130, 6266, 7664, 9312, 11332, 13723, 16603, 20004, 24112, 28942, 34708, 41522, 49612, 59031, 70308, 83479, 98992

Offset: 0

Clark Kimberling, Feb 06 2014

nonn

a(n) + A237365(n) = A000041(n).

Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - Gus Wiseman, Mar 18 2023

			Among the 22 partitions of 8, these qualify:  [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..800

These partitions have ranks A361204.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts, reverse A058398.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A000975, A027193, A114638, A240219, A325347, A360005, A360244.

z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (*A237363*)
Table[PartitionsP[n] - t[[n]], {n, 1, z}] (*A237365*) (* Peter J. C. Moses, Feb 06 2014 *)
Table[Length[Select[IntegerPartitions[n],Median[Differences[#]]==0&]],{n,0,30}] (* Gus Wiseman, Mar 18 2023 *)

A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12

Offset: 0

Gus Wiseman, Aug 27 2023

Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.

			Triangle begins:
  1
  1  1
  1  2  1
  1  3  1  2
  1  4  2  3  2
  1  5  2  5  3  3
  1  6  3  8  4  4  4
  1  7  3 11  6  6  6  5
  1  8  4 14  9  8 10  7  6
  1  9  4 19 11 11 14 11  9  8
  1 10  5 23 14 15 21 15 14 11 10
  1 11  5 28 17 19 28 22 20 17 15 12
  1 12  6 34 21 22 40 28 28 24 24 17 15
  1 13  6 40 25 27 50 38 37 34 35 27 22 18
  1 14  7 46 29 32 65 49 50 43 51 38 35 26 22
  1 15  7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
    .    1           2     3         4       5
         1+1         2+2   1+2       1+3     1+4
         1+1+1             1+1+2     1+1+3   2+3
         1+1+1+1           1+1+1+2
         1+1+1+1+1         1+2+2
Row n = 5 counts the following positive linear combinations:
  .  1*1  1*2  1*3      1*4      1*5
     2*1  2*2  1*2+1*1  1*3+1*1  1*3+1*2
     3*1       1*2+2*1  1*3+2*1  1*4+1*1
     4*1       1*2+3*1
     5*1       2*2+1*1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column n = k is A000009.
Column k = 0 is A000012.
Column k = 1 is A000027.
Row sums are A000070.
Column k = 2 is A008619.
Columns are partial sums of columns of A116861.
Column k = 3 appears to be the partial sums of A137719.
Diagonal n = 2k is A364910.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A114638 counts partitions where (length) = (sum of distinct parts).
A116608 counts partitions by number of distinct parts.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.

Table[Length[Select[Array[IntegerPartitions,n+1,0,Join],Total[Union[#]]==k&]],{n,0,9},{k,0,n}]

T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - Andrew Howroyd, Jan 11 2024

cp's OEIS Frontend

A376311 Position of first appearance of n in the sequence of first differences of squarefree numbers, or the sequence ends if there is none.

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A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

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A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

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A361849 Number of integer partitions of n such that the maximum is twice the median.

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A056556 First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.

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A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

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A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

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