A261784 -id:A261784 - cp's OEIS frontend

A261781 Number T(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 2, 3, 0, 4, 16, 13, 0, 8, 66, 132, 75, 0, 16, 248, 924, 1232, 541, 0, 32, 892, 5546, 13064, 13060, 4683, 0, 64, 3136, 30720, 114032, 195020, 155928, 47293, 0, 128, 10888, 162396, 893490, 2327960, 3116220, 2075948, 545835

Offset: 0

Alois P. Heinz, Aug 31 2015

From Vaclav Kotesovec, Oct 14 2017: (Start)
Conjecture: For k > 0 the recurrence order for column k is equal to k*(k+1)/2.
Column k > 0 is asymptotic to c(k) * d(k)^n, where c(k) and d(k) are constants (dependent only on k).
k                           c(k)                            d(k)
1  A131577(n) ~ 0.50000000000000000000000000 * 2.00000000000000000000000000^n.
2  A293579(n) ~ 0.60355339059327376220042218 * 3.41421356237309504880168872^n.
3  A293580(n) ~ 0.64122035031051210658648604 * 4.84732210186307263951891624^n.
4  A293581(n) ~ 0.66065168848540565019767995 * 6.28521350788324520158143964^n.
5  A293582(n) ~ 0.67250239588725756267924287 * 7.72502395887257562679242875^n.
6  A293583(n) ~ 0.68048292906885160660288253 * 9.16579514882621927923459043^n.
7  A293584(n) ~ 0.68622254929933439577377124 * 10.6071156901906815408327973^n.
8  A293585(n) ~ 0.69054873168854973836384871 * 12.0487797070167958138215794^n.
9  A293586(n) ~ 0.69392626461456654033893782 * 13.4906727630621977261008808^n.
10 A293587(n) ~ 0.69663630864564830007443110 * 14.9327261729129660014886221^n.
---
Conjecture: d(k+1) - d(k) tends to 1/log(2).
d(2) - d(1) = 1.414213562373095048801688724209698...
d(3) - d(2) = 1.433108539489977590717227522340838...
d(4) - d(3) = 1.437891406020172562062523400686067...
d(5) - d(4) = 1.439810450989330425210989107036901...
d(6) - d(5) = 1.440771189953643652442161677346934...
d(7) - d(6) = 1.441320541364462261598206961226199...
d(8) - d(7) = 1.441664016826114272988782079622148...
d(9) - d(8) = 1.441893056045401912279301345910755...
d(10)- d(9) = 1.442053409850768275387741352145193...
1 / log(2)  = 1.442695040888963407359924681001892...
(End)

			A(3,2) = 16: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,    3;
  0,  4,   16,    13;
  0,  8,   66,   132,     75;
  0, 16,  248,   924,   1232,    541;
  0, 32,  892,  5546,  13064,  13060,   4683;
  0, 64, 3136, 30720, 114032, 195020, 155928, 47293;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Table 2.

Columns k=0..10 give A000007, A131577, A293579, A293580, A293581, A293582, A293583, A293584, A293585, A293586, A293587.
Row sums give A120733.
Main diagonal gives A000670.
T(2n,n) gives A261784.
T(n+1,n)/2 gives A083385.
Cf. A261719 (same for partitions), A261780.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n==0, 1,
    Sum[A[n-j, k]*Binomial[j+k-1, k-1], {j, 1, n}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261780(n,k-i).

A261828 Number of compositions of 2n into distinct parts where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur at least once in the composition.

Original entry on oeis.org

1, 1, 15, 832, 14791, 2008546, 55380132, 2868333476, 511805155863, 31512728488918, 2638310862477610, 926651539894899446, 74254761492776175196, 6851495812540548188072, 9541620342114654822145972, 611287722968440282212322702, 58354641005988089624088037623

Offset: 0

Alois P. Heinz, Sep 02 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A261784, A261836.

b:= proc(n, i, p, k) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> add(b(2*n$2, 0, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..20);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; a[n_] := Sum[b[2*n, 2*n, 0, n-i]*(-1)^i*Binomial[n, i], {i, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2017, translated from Maple *)

a(n) = A261836(2n,n).

cp's OEIS Frontend

A261781 Number T(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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