A238341 -id:A238341 - cp's OEIS frontend

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25

Offset: 0

Gus Wiseman, May 04 2023

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  11   3   1
   0  16   4   2
   0  21   6   3
   0  29   8   4   1
   0  43   7   5   1
   0  54  13   8   2
   0  78  12   8   3
   0 102  17  11   5
   0 131  26  12   6   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1, 22, 4, 2, 1, 0, 0, 0, 0, 1, 30, 7, 2, 1, 1, 0, 0, 0, 0, 1, 42, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Clark Kimberling

At least one part is k and each part is at least k.
From Emeric Deutsch, Feb 19 2006: (Start)
Also number of partitions of n in which the largest part occurs exactly k times. Example: T(6,2)=2 because we have [3,3] and [2,2,1,1].
G.f. of column k is x^k/prod(j>=k, 1-x^j ) (k>=1).
Row sums yield the partition numbers (A000041).
T(n,1) = A000041(n-1) (the partition numbers).
T(n,2) = A002865(n-2) (n>=2).
T(n,3)=A026796(n). T(n,4) = A026797(n). T(n,5) = A026798(n). T(n,6) = A026799(n). T(n,7) = A026800(n). T(n,8) = A026801(n). T(n,9) = A026802(n). T(n,10) = A026803(n).
Sum(k*T(n,k),k=1..n) = A046746(n). (End)
Triangle inverse = A161363. - Gary W. Adamson, Jun 07 2009
T(n,g) is also the number of not necessarily connected 2-regular graphs with girth exactly g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012
From Bob Selcoe, Jul 24 2014 (Start):
Below is a process to generate equations for column k.
Let P be the partition numbers A000041(n-j) and let f(k) denote equations which generate column k.
To find f(k), start with f(1) = P(n-j), j=1. Thus T(n,1) = f(1) = P(n-1). This is the equation for column 1.
To find f(k) k>1, first sum the terms of f(k-1) replacing the value j with j+1, and then subtract the terms of f(k-1) replacing the value j with j+k. So to find f(2) (i.e., the equation for column 2, where k=2), start with f(1) = P(n-1); first replace j with j+1 (yielding P(n-2)), and then replace j with j+2 (yielding P(n-3)).  Subtracting the second term from the first, we get: f(2) = P(n-2) - P(n-3).
To find f(3), start with f(2), replace j with j+1 (yielding P(n-3) - P(n-4)) and then replace j with j+3 (yielding P(n-5) - P(n-6)). Subtracting the second group of terms from the first, we get: f(3) = P(n-3) - P(n-4) - P(n-5) + P(n-6). This is the equation for column 3; also the equation for T(n,3) = A026796(n). So for example, T(13,3) = 5 because P(13-3) - P(13-4) - P(13-5) + P(13-6) = 42 - 30 - 22 + 15 = 5.
Continue as above to find f(k) k={4..inf.}.  This will generate equations for T(n,4) = A026797(n), T(n,5) = A026798(n), T(n,6) = A026799(n), ad inf.
(End)

			T(12,3) = 4 because we have [9,3], [6,3,3], [5,4,3] and [3,3,3,3]. - Edited by _Bob Selcoe_, Sep 03 2016
Triangle starts:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    5,  1, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   11,  2, 1, 0, 0, 0, 1;
   15,  4, 1, 1, 0, 0, 0, 1;
   22,  4, 2, 1, 0, 0, 0, 0, 1;
   30,  7, 2, 1, 1, 0, 0, 0, 0, 1;
   42,  8, 3, 1, 1, 0, 0, 0, 0, 0, 1;
   56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
   77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  135, 24, 9, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Kevin Brown, On Euler's Pentagonal Theorem, 1994-2008.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g.
Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No. 1, December 2008. pp. 176-187.
Tilman Piesk, First 50 rows and illustrations of columns n = 2, 3, 4, 5, 6, 7, 8. a(n, k) is the number of gray fields in row n of table k.

Row sums give A000041.
Cf. A000041, A046746, A008284, A161363, A161364, A238341.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: this sequence (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012

g:=sum(t^i*x^i/product(1-x^j,j=i..30),i=1..30): gser:=simplify(series(g,x=0,19)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # Emeric Deutsch, Feb 19 2006
nmax:=13; for n from 1 to nmax do T(n,n):=1 od: for n from 1 to nmax do for k from floor(n/2)+1 to n-1 do T(n,k):=0 od: od: for n from 2 to nmax do for k from 1 to floor(n/2) do T(n,k):=sum(T(n-k,i),i=k..n-k) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
nmax:=13; with(combinat): for n from 1 to nmax do for k from n+1 to nmax do T(n,k):=0 od: od: for n from 1 to nmax do T(n,1):=numbpart(n-1) od: for n from 1 to nmax do T(n,n):=1 od: for n from 2 to nmax do for k from 2 to n-1 do T(n,k) := T(n-1,k-1) - T(n-k,k-1) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
#
p:= (f, g)-> zip((x,y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local h;
      h:= `if`(n=i and i>0, [0$(i-1), 1], []);
      `if`(i<1, h, p(p(h, b(n, i-1)), `if`(n b(n, n)[]:
seq(T(n), n=1..14); # Alois P. Heinz, Mar 28 2012

t[n_, k_] /; k<1 || k>n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[t[n-k, i], {i, k, n-k}]; Flatten[ Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, May 11 2012, after PARI *)

{T(n, k) = if( k<1 || k>n, 0, if( n==k, 1, sum(i=k, n-k, T(n-k, i))))} \\ Michael Somos, Feb 06 2003

A026794(n,k)=#select(p->p[1]==k,partitions(n,[k,n])) \\ For illustration: Creates the list of all partitions of n with smallest part equal to k. - M. F. Hasler, Jun 14 2018

T(n, k) = sum{T(n-k, i), k<=i<=n-k} for k=1, 2, ..., m, T(n, k)=0 for k=m+1, ..., n-1, where m=floor(n/2); T(n, n)=1 for n >= 1.
G.f.: G(t,x)=sum(t^i*x^i/product(1-x^j, j=i..infinity), i=1..infinity). - Emeric Deutsch, Feb 19 2006
G.f.: Sum_{k>=1} tx^k/(1-tx^k)/product(1-x^j,j=1..k-1). - Emeric Deutsch, Mar 13 2006
T(n,k) = T(n-1,k-1) - T(n-k,k-1) for n>=2 and 2<=k<=(n-1) with T(n,1) = A000041(n-1), T(n,n) = 1 for n>=1 and T(n,k) = 0 for k>n. - Johannes W. Meijer, Jun 21 2010
T(k,k) = 1 and T(n,1) = row sum (n-1); thus Meijer's 2010 formula generates the triangle without a priori reference to A000041 (the partition sequence). - Bob Selcoe, Sep 03 2016

More terms from Emeric Deutsch, Feb 19 2006

A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 10, 4, 1, 0, 13, 7, 2, 0, 16, 11, 3, 0, 23, 14, 4, 1, 0, 30, 19, 6, 1, 0, 35, 29, 11, 2, 0, 50, 34, 14, 3, 0, 61, 46, 23, 5, 0, 73, 69, 27, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2

Offset: 0

Gus Wiseman, May 04 2023

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  10   4   1
   0  13   7   2
   0  16  11   3
   0  23  14   4   1
   0  30  19   6   1
   0  35  29  11   2
   0  50  34  14   3
   0  61  46  23   5
   0  73  69  27   6   1
   0  95  81  44  10   1
Row n = 8 counts the following partitions:
  (8)         (53)     (431)
  (44)        (62)     (521)
  (332)       (71)
  (422)       (3221)
  (611)       (3311)
  (2222)      (4211)
  (5111)      (32111)
  (22211)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.
Column k = 1 is A362610, ranks A359178.
This statistic (co-mode count) is ranked by A362613.
For mode instead of co-mode we have A362614, ranked by A362611.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - Alois P. Heinz, May 07 2024

A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 8, 3, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 20, 18, 14, 5, 6, 0, 1, 0, 34, 35, 24, 21, 6, 7, 0, 1, 0, 59, 60, 59, 35, 27, 7, 8, 0, 1, 0, 96, 121, 108, 85, 49, 35, 8, 9, 0, 1, 0, 167, 217, 213, 175, 125, 63, 44, 9, 10, 0, 1, 0, 282, 391, 419, 366, 258, 176, 80, 54, 10, 11, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Conjecture: Generally, for k > 0 is a(n) ~ n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). Holds for all k<=10. - Vaclav Kotesovec, May 02 2014
G.f.: 1 + Sum_{i>0} (-y*(x^i)*(x - 1)^2)/( (x^(i+1) + x - 1)*((x^i)*(x*(y - 1) - y) - x + 1) ). - John Tyler Rascoe, Oct 15 2024
Sum_{k=0..n} k * T(n,k) = A097941(n). - Alois P. Heinz, Oct 15 2024

			Triangle starts:
00:  1;
01:  0,    1;
02:  0,    1,    1;
03:  0,    3,    0,    1;
04:  0,    3,    4,    0,    1;
05:  0,    8,    3,    4,    0,    1;
06:  0,   11,   10,    5,    5,    0,    1;
07:  0,   20,   18,   14,    5,    6,    0,    1;
08:  0,   34,   35,   24,   21,    6,    7,    0,   1;
09:  0,   59,   60,   59,   35,   27,    7,    8,   0,   1;
10:  0,   96,  121,  108,   85,   49,   35,    8,   9,   0,   1;
11:  0,  167,  217,  213,  175,  125,   63,   44,   9,  10,   0,  1;
12:  0,  282,  391,  419,  366,  258,  176,   80,  54,  10,  11,  0,  1;
13:  0,  475,  709,  808,  730,  579,  371,  236,  99,  65,  11, 12,  0,  1;
14:  0,  800, 1281, 1522, 1481, 1202,  861,  513, 309, 120,  77, 12, 13,  0, 1;
15:  0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A238341 (the same for largest part).
Columns k=0-10 give: A000007, A105039, A241862, A241863, A241864, A241865, A241866, A241867, A241868, A241869, A241870.
Row sums are A011782.
T(2*n,n) gives A232665(n).
Cf. A097941.

b:= proc(n, s) option remember;`if`(n=0, 1,
      `if`(n`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
     binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
seq(seq(T(n, k), k=0..n), n=0..15);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Nov 07 2014, translated from Maple *)

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=1+sum(i=1,N,(-y*(x^i)*(x-1)^2)/((x^(i+1)+x-1)*((x^i)*(x*(y-1)-y)-x+1)))); for(i=0,N-1, print(Vecrev(polcoef(h,i))))}
T_xy(15) \\ John Tyler Rascoe, Oct 15 2024

A097979 Total number of largest parts in all compositions of n.

Original entry on oeis.org

1, 3, 6, 12, 23, 46, 91, 183, 367, 737, 1478, 2962, 5928, 11858, 23707, 47384, 94698, 189260, 378277, 756160, 1511730, 3022672, 6044472, 12088395, 24177600, 48359695, 96732370, 193495606, 387057584, 774248858, 1548754115, 3097980230, 6196797193, 12395022288

Offset: 1

Vladeta Jovovic, Sep 07 2004

Also number of compositions of n+1 with unique largest part. - Vladeta Jovovic, Apr 03 2005

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Vincenzo Librandi)

Cf. A097941, A046746.

Column k=1 of A238341.

nn=32; Drop[CoefficientList[Series[Sum[x^j/(1 - (x - x^(j + 1))/(1 - x))^2, {j, 1, nn}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Mar 31 2014 *)
b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_, k_] := Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]; a[0, 0] = 1; a[, 0] = 0; a[n] := a[n+1, 1]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 10 2015, after A238341 *)

{ b(t)=local(r);sum(k=1,t, forstep(s=t%k,t-k,k,u=(t-k-s)\k;r+=binomial(-2,s)*(-2)^(s-u)*binomial(s,u)));r }
{ a(n)=b(n)-2*b(n-1)+b(n-2) } \\ Max Alekseyev, Apr 16 2005

G.f.: (1-x)^2 * Sum_{k>=1} x^k/(1-2*x+x^(k+1))^2.

a(n) ~ 2^(n-1)/log(2). - Vaclav Kotesovec, Apr 30 2014

More terms from Max Alekseyev, Apr 16 2005

A243737 Number of compositions of n with exactly two occurrences of the largest part.

Original entry on oeis.org

1, 0, 1, 3, 7, 13, 25, 46, 89, 175, 351, 710, 1443, 2926, 5920, 11936, 23987, 48072, 96139, 191977, 382992, 763686, 1522581, 3035979, 6055454, 12082887, 24120923, 48174935, 96259627, 192418152, 384772810, 769651514, 1539889604, 3081525905, 6167365392

Offset: 2

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 2..650

Column k=2 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=2;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=2..40);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=2; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]) ; Table[ a[n], {n, 2, 40}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A363224 Number of integer compositions of n in which the least part appears more than once.

Original entry on oeis.org

0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.

			The a(1) = 0 through a(6) = 21 compositions:
  .  (11)  (111)  (22)    (113)    (33)
                  (112)   (131)    (114)
                  (121)   (311)    (141)
                  (211)   (1112)   (222)
                  (1111)  (1121)   (411)
                          (1211)   (1113)
                          (2111)   (1122)
                          (11111)  (1131)
                                   (1212)
                                   (1221)
                                   (1311)
                                   (2112)
                                   (2121)
                                   (2211)
                                   (3111)
                                   (11112)
                                   (11121)
                                   (11211)
                                   (12111)
                                   (21111)
                                   (111111)

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

The complement is counted by A105039.
For partitions instead of compositions we have A117989.
Row sums of columns k > 1 of A238342.
If all parts appear more than once we have A240085, for partitions A007690.
If the least part appears exactly twice we have A241862.
For greatest instead of least we have A363262, see triangle A238341.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
Cf. A002865, A008284, A097979, A243737.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Min@@#]>1&]],{n,15}]

C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024

G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024

A363262 Number of integer compositions of n in which the greatest part appears more than once.

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 18, 37, 73, 145, 287, 570, 1134, 2264, 4526, 9061, 18152, 36374, 72884, 146011, 292416, 585422, 1171632, 2344136, 4688821, 9376832, 18749169, 37485358, 74939850, 149813328, 299492966, 598729533, 1196987066, 2393137399, 4784846896, 9567357951

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.

			The a(2) = 1 through a(6) = 9 compositions:
  (11)  (111)  (22)    (122)    (33)
               (1111)  (212)    (222)
                       (221)    (1122)
                       (11111)  (1212)
                                (1221)
                                (2112)
                                (2121)
                                (2211)
                                (111111)

For partitions instead of compositions we have A002865.
The complement is counted by A097979 shifted left.
Row sums of columns k > 1 of A238341.
If all parts appear more than once we have A240085, for partitions A007690.
If the greatest part appears exactly twice we have A243737.
For least instead of greatest we have A363224, see triangle A238342.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
Cf. A008284, A105039, A117989, A362607, A362608, A362612, A362614.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Max@@#]>1&]],{n,15}]

cp's OEIS Frontend

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

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A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

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A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

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A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

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A097979 Total number of largest parts in all compositions of n.

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A243737 Number of compositions of n with exactly two occurrences of the largest part.

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A363224 Number of integer compositions of n in which the least part appears more than once.

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