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.

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

Offset: 0

Author

Ji Young Choi, Dec 16 2019

Keywords

• nonn
• tabl
• base

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.

Examples

			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.
		

Links

• J. Y. Choi, Indispensable digits for digit sums, Notes Number Theory Discrete Math 25 (2019), pp. 40-48.
• J. Y. Choi, Digit sums generalizing binomial coefficients, J. Integer Seq. 22 (2019), Article 19.8.3.

Crossrefs

Cf. A008287, A330381.

Programs

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 *)

PARI

A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
T(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);

Formula

T(n, k) = A008287(n, 3k-2) + A008287(n, 3k-1) + A008287(n, 3k).