|
|
A330509
|
|
Triangle read by rows: T(n,k) is the number of 4-ary strings of length n with k indispensable digits, with 0 <= k <= n.
|
|
2
|
|
|
1, 1, 3, 1, 9, 6, 1, 19, 34, 10, 1, 34, 115, 91, 15, 1, 55, 301, 445, 201, 21, 1, 83, 672, 1582, 1338, 392, 28, 1, 119, 1344, 4600, 6174, 3410, 700, 36, 1, 164, 2478, 11623, 22548, 19784, 7723, 1170, 45, 1, 219, 4290, 26452, 69834, 88428, 55009, 15999, 1857, 55
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
A digit in a string is called indispensable if it is greater than the following digit or equal to the following digits which are eventually greater than the following digit. We also assume that there is an invisible digit 0 at the end of any string. For example, in 7233355548, the digits 7, 5, 5, 5, and 8 are indispensable.
T(n, k) is also the number of integers m where the length of the base-4 representation of m is n and the digit sum of the base-4 representation of 3m is 3k.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Triangle begins
1;
1, 3;
1, 9, 6;
1, 19, 34, 10;
1, 34, 115, 91, 15;
1, 55, 301, 445, 201, 21;
...
There is 1 string (00) of length 2 with 0 indispensable digits.
There are 9 strings (01, 02, 03, 10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digit.
There are 6 strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits.
Hence T(2,0)=1, T(2,1)=9, T(2,2)=6.
|
|
MATHEMATICA
|
Table[Total@ Map[Sum[Binomial[n, i] Binomial[n, # - 2 i], {i, 0, #/2}] &, 3 k + {-2, -1, 0}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 23 2019, after Jean-François Alcover at A008287 *)
|
|
PROG
|
(PARI) A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|