|
|
A181695
|
|
Triangle read by rows: T(n,m) = number of solutions x_1 + x_2 + ... + x_k <= n, where 1 <= x_i <= m, and any k >= 1.
|
|
1
|
|
|
1, 2, 3, 3, 6, 7, 4, 11, 14, 15, 5, 19, 27, 30, 31, 6, 32, 51, 59, 62, 63, 7, 53, 95, 115, 123, 126, 127, 8, 87, 176, 223, 243, 251, 254, 255, 9, 142, 325, 431, 479, 499, 507, 510, 511, 10, 231, 599, 832, 943, 991, 1011, 1019, 1022, 1023, 11, 375, 1103, 1605
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
For a fixed m, generating function is 1/(1-2*x+x^(m+1)) - 1/(1-x).
T(n,m) = Sum_{i=0..floor(n/(m+1))} binomial(n-mi, i)*(-1)^i*2^(n-(m+1)i) - 1.
T(n,m) = 2^m - 1 + Sum_{j=m+1..n} A126198(j,m).
|
|
EXAMPLE
|
Triangle begins:
1;
2, 3;
3, 6, 7;
4, 11, 14, 15;
5, 19, 27, 30, 31;
6, 32, 51, 59, 62, 63;
7, 53, 95, 115, 123, 126, 127;
...
Could also be extended to a square array:
1, 1, 1, 1, 1, 1, 1, ...
2, 3, 3, 3, 3, 3, 3, ...
3, 6, 7, 7, 7, 7, 7, ...
4, 11, 14, 15, 15, 15, 15, ...
5, 19, 27, 30, 31, 31, 31, ...
6, 32, 51, 59, 62, 63, 63, ...
7, 53, 95, 115, 123, 126, 127, ...
|
|
PROG
|
(PARI) { T(n, m) = sum(i=0, n\(m+1), binomial(n-m*i, i) * (-1)^i * 2^(n-(m+1)*i) ) - 1 }
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|