A330381 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.

1, 1, 2, 1, 5, 3, 1, 9, 13, 4, 1, 14, 35, 26, 5, 1, 20, 75, 96, 45, 6, 1, 27, 140, 267, 216, 71, 7, 1, 35, 238, 623, 750, 427, 105, 8, 1, 44, 378, 1288, 2123, 1800, 770, 148, 9, 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10, 1, 65, 825, 4302, 11505, 17303

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 ternary representation of m is less than or equal to n and the digit sum of the ternary representation of 2m is 2k.

Links

Table of n, a(n) for n=0..60.

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.

Formula

T(n, k) = A027907(n, 2k-1) + A027907(n, 2k).

Example

Triangle begins
  1;
  1,  2;
  1,  5,  3;
  1,  9, 13,  4;
  1, 14, 35, 26,  5;
  1, 20, 75, 96, 45, 6;
  ...
There is 1 string (00) of length 2 with 0 indispensable digits.
There are 5 strings (01, 02, 10, 20, 12) of length 2 with 1 indispensable digit.
There are 3 strings (11, 21, 22) of length 2 with 2 indispensable digits.
Hence T(2, 0) = 1, T(2, 1) = 5, T(2, 2) = 3.

Mathematica

Table[Total@ Map[Sum[Binomial[n, i] Binomial[n - i, # - 2 i], {i, 0, n}] &, 2 k + {-1, 0}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 23 2019, after Adi Dani at A027907 *)

Prog

(PARI) A027907(n, k) = if(n<0, 0, polcoeff((1 + x + x^2)^n, k));
T(n, k) = A027907(n, 2*k-1) + A027907(n, 2*k); \\ Jinyuan Wang, Dec 14 2019

Crossrefs

Cf. A027907, A330509.

Sequence in context: A240192 A264751 A209130 * A210792 A105728 A120095

Adjacent sequences: A330378 A330379 A330380 * A330382 A330383 A330384

Keyword

nonn,base,tabl

Author

Ji Young Choi, Dec 12 2019

Extensions

More terms from Jinyuan Wang, Dec 14 2019

Status

approved