A097941 -id:A097941 - cp's OEIS frontend

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.

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

A097939 Sum of the smallest parts of all compositions of n.

Original entry on oeis.org

1, 3, 6, 12, 22, 42, 79, 151, 291, 566, 1106, 2175, 4293, 8499, 16864, 33523, 66727, 132958, 265137, 529050, 1056169, 2109282, 4213710, 8419697, 16827079, 33634489, 67237513, 134424624, 268768414, 537407062, 1074605619, 2148875961, 4297212424, 8593556211, 17185713097, 34369170909

Offset: 1

Vladeta Jovovic, Sep 05 2004

Sums of the antidiagonals of A099238. - Paul Barry, Oct 08 2004

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (correcting an earlier b-file from Vincenzo Librandi)

Cf. A046746, A092309, A097940, A097941, A099238, A336902, A336903.

Column k=1 of A322427.

A097939:=n->add(add(binomial(n-r*(k+1)-1,k), k=0..floor((n-r-1)/(r+1))), r=0..n-1): seq(A097939(n), n=1..50); # Wesley Ivan Hurt, Dec 03 2016
# second Maple Program:
b:= proc(n, m) option remember; `if`(n=0, m,
      add(b(n-j, min(j, m)), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 26 2020

Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

N=66; x='x+O('x^N);
gf= sum(k=1,N, x^k/(1-x-x^k) );
Vec(gf)
/* Joerg Arndt, Jan 01 2013 */

{a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,1/(1-x +x*O(x^n))^d) ),n)}

G.f.: Sum_{k>=1} x^k/(1-x-x^k).
a(n) = Sum_{r=0..n-1} Sum_{k=0..floor((n-r-1)/(r+1))} binomial(n-r(k+1)-1, k). - Paul Barry, Oct 08 2004
G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/((x^k+x-1)*(x^(k+1)+x-1)). - Vladeta Jovovic, Apr 23 2006
G.f.: Sum_{k>=1} x^k/((1-x)^k*(1-x^k)). - Vladeta Jovovic, Mar 02 2008
G.f.: Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series) where a=1/(1-x). - Joerg Arndt, Jan 30 2011
G.f.: Sum_{n>=1} (a*x)^n/(1-x^n) where a=1/(1-x). - Joerg Arndt, Jan 01 2013
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1-x)^d. - Paul D. Hanna, Jul 18 2013
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 28 2014

More terms from Robert G. Wilson v, Sep 08 2004

A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 46, 107, 244, 547, 1213, 2665, 5807, 12567, 27042, 57899, 123428, 262115, 554750, 1170538, 2463154, 5170462, 10829234, 22635087, 47223412, 98353299, 204519549, 424665001, 880581806, 1823667221, 3772341661, 7794697759, 16089424392, 33178906531, 68357928558

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of squarefree parts in all compositions (ordered partitions) of n.

			a(4) = 19 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 0 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

Alois P. Heinz, Table of n, a(n) for n = 1..3312
Index entries for sequences related to compositions

Cf. A005117, A008683, A011782, A059570, A097941, A097979, A102291, A281573, A284943.

a:= proc(n) option remember; add(`if`(numtheory[
      issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k (1 - x)^2/(1 - 2 x)^2, {k, 1, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); Vec(sum(k=1, 34, moebius(k) ^2*x^k*(1 - x)^2/(1 - 2*x)^2)) \\ Indranil Ghosh, Apr 06 2017

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

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.

Original entry on oeis.org

0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.

			a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.

Alois P. Heinz, Table of n, a(n) for n = 1..3313
Index entries for sequences related to compositions

Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.

b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017

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

A308630 Triangle T(n,k) read by rows: the sum of all smallest parts among all k-compositions of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 6, 9, 12, 5, 6, 12, 18, 24, 20, 6, 7, 12, 27, 40, 50, 30, 7, 8, 20, 36, 68, 100, 90, 42, 8, 9, 20, 54, 108, 175, 210, 147, 56, 9, 10, 30, 72, 160, 290, 420, 392, 224, 72, 10, 11, 30, 90, 224, 460, 756, 882, 672, 324, 90, 11, 12, 42, 120, 312, 700, 1272, 1764, 1680, 1080, 450, 110

Offset: 1

R. J. Mathar, Jun 12 2019

			The triangle starts in row n=1 with columns 1<=k<=n as:
   1;
   2,  2;
   3,  2,  3;
   4,  6,  6,  4;
   5,  6,  9, 12,  5;
   6, 12, 18, 24, 20,  6;
   7, 12, 27, 40, 50, 30,  7;
   8, 20, 36, 68,100, 90, 42,  8;
   9, 20, 54,108,175,210,147, 56,  9;
  10, 30, 72,160,290,420,392,224, 72, 10;
  ...

Knopfmacher, Arnold; Munagi, Augustine O. Smallest parts in compositions, Kotsireas, Ilias S. (ed.) et al., Advances in combinatorics. 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26-29, 2011. Berlin: Springer. 197-207 (2013).

Cf. A097941 (number of smallest parts), A002378 (k=2), A144677 (column k=3 divided by 3), A097940 (row sums).

A308630 := proc(n,k)
    add(j*binomial(n-(j-1)*k-2,k-2),j=1..floor(n/k)) ;
    %*k ;
end proc:

T(n,k) = k*sum_{j=1..floor(n/k)} binomial(n-(j-1)*k-2, k-2).

cp's OEIS Frontend

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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A097939 Sum of the smallest parts of all compositions of n.

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A284942 Expansion of Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

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A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

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A308630 Triangle T(n,k) read by rows: the sum of all smallest parts among all k-compositions of n.

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Maple

Formula

A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.