OFFSET
0,5
COMMENTS
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)
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
E. Munarini, M. Poneti, S. Rimaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Table 2.
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261780(n,k-i).
EXAMPLE
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;
...
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 31 2015
STATUS
approved