A097941 -id:A097941

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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.

+10
18

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

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

LINKS

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

EXAMPLE

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;

...

MAPLE

b:= proc(n, s) option remember; `if`(n=0, 1,

`if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))

end:

T:= (n, k)->`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);

MATHEMATICA

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i+k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 07 2014, translated from Maple *)

PROG

(PARI)

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

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 25 2014

STATUS

approved

A097979

Total number of largest parts in all compositions of n.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) { 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

CROSSREFS

Cf. A097941, A046746.

Column k=1 of A238341.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 07 2004

EXTENSIONS

More terms from Max Alekseyev, Apr 16 2005

STATUS

approved

A097939

Sum of the smallest parts of all compositions of n.

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI)

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 */

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

CROSSREFS

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

Column k=1 of A322427.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 05 2004

EXTENSIONS

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

STATUS

approved

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).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..3312

Index entries for sequences related to compositions

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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]]

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 06 2017

STATUS

approved

A284943

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..3313

Index entries for sequences related to compositions

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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]]

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 06 2017

STATUS

approved

A308630

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..77.

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).

FORMULA

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

EXAMPLE

The triangle starts in row n=1 with columns 1<=k<=n as:

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;

...

MAPLE

A308630 := proc(n, k)

add(j*binomial(n-(j-1)*k-2, k-2), j=1..floor(n/k)) ;

%*k ;

end proc:

CROSSREFS

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

KEYWORD

nonn,easy,tabl

AUTHOR

R. J. Mathar, Jun 12 2019

STATUS

approved

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