A330510 Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.

2, 3, 3, 4, 10, 4, 5, 22, 22, 5, 6, 40, 70, 40, 6, 7, 65, 171, 171, 65, 7, 8, 98, 356, 534, 356, 98, 8, 9, 140, 665, 1373, 1373, 665, 140, 9, 10, 192, 1148, 3088, 4246, 3088, 1148, 192, 10, 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11

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 ternary representation of m is n+k and the digit sum of the ternary representation of 2m is 2(k+1).

Examples

			Triangle begins
  2;
  3,   3;
  4,  10,   4;
  5,  22,  22,   5;
  6,  40,  70,  40,   6;
  7,  65, 171, 171,  65,   7;
  ...
There are 4 strings (100, 112, 120, 200) of length 3 with 1 indispensable digits and a nonzero leading digit.
There are 10 strings (101, 102, 110, 121, 122, 201, 202, 210, 212, 220) of length 3 with 2 indispensable digits are a nonzero leading digit.
There are 4 strings (111, 211, 221, 222) of length 3 with 3 indispensable digits and a nonzero leading digit.
Hence T(2,0)=4, T(2,1)=10, T(2,2)=4.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• 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. A005773, A330381, A027907.

Programs

PARI

A027907(n, k) = if(n<0, 0, polcoef((1 + x + x^2)^n, k));
T(n,k) = {A027907(n+1, 2*k+1) + A027907(n+1, 2*k+2) - A027907(n, 2*k+1) - A027907(n, 2*k+2)} \\ Andrew Howroyd, Dec 20 2019

Formula

T(n, k) = A330381(n+1, k+1) - A330381(n, k+1).