|
|
A126198
|
|
Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.
|
|
9
|
|
|
1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 8, 13, 15, 16, 1, 13, 24, 29, 31, 32, 1, 21, 44, 56, 61, 63, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k (see A048004). - N. J. A. Sloane, Apr 03 2011
Higher Order Fibonacci numbers: A126198(n,k) = Sum_{h=0..k} A048004(n,h); for example, A126198(7,3) = Sum_{h=0..3} A048004(7,h) or A126198(7,3) = 1 + 33 + 47 + 27 = 108, the 7th tetranacci number. A048004 row(7) produces A126198 row(7) list of 1,34,81,108,120,125,127,128 which are 1, the 7th Fibonacci, the 7th tribonacci, ... 7th octanacci numbers. - Richard Southern, Aug 04 2017
|
|
REFERENCES
|
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.
|
|
LINKS
|
|
|
FORMULA
|
G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
|
|
EXAMPLE
|
Triangle begins:
1;
1, 2;
1, 3, 4;
1, 5, 7, 8;
1, 8, 13, 15, 16;
1, 13, 24, 29, 31, 32;
1, 21, 44, 56, 61, 63, 64;
Could also be extended to a square array:
1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 ...
1 3 4 4 4 4 4 ...
1 5 7 8 8 8 8 ...
1 8 13 15 16 16 16 ...
1 13 24 29 31 32 32 ...
1 21 44 56 61 63 64 ...
which when read by antidiagonals (downwards) gives A048887.
|
|
MAPLE
|
A126198 := proc(n, k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)), x=0, n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ", A126198(n, k)); od; od; # R. J. Mathar, Mar 09 2007
# second Maple program:
T:= proc(n, k) option remember;
if n=0 or k=1 then 1
else add(T(n-j, k), j=1..min(n, k))
fi
end:
|
|
MATHEMATICA
|
rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011, after Max Alekseyev *)
|
|
CROSSREFS
|
2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|