A330381 -id:A330381 - cp's OEIS frontend

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

Ji Young Choi, Dec 16 2019

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.

			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.

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.

Cf. A008287, A330381.

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 *)

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);

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

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.

Original entry on oeis.org

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

Ji Young Choi, Dec 16 2019

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).

			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.

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.

Cf. A005773, A330381, A027907.

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

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

A340620 T(n,k) is the number of 4-ary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit with 0 <= k <= n.

Original entry on oeis.org

3, 6, 6, 10, 28, 10, 15, 81, 81, 15, 21, 186, 354, 186, 21, 28, 371, 1137, 1137, 371, 28, 36, 672, 3018, 4836, 3018, 672, 36, 45, 1134, 7023, 16374, 16374, 7023, 1134, 45, 55, 1812, 14829, 47286, 68644, 47286, 14829, 1812, 55, 66, 2772, 29043, 121314, 240021, 240021, 121314, 29043, 2772, 66

Offset: 0

Ji Young Choi, Jan 13 2021

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 the string 33102232, the digits 3, 3, 1, 3, and 2 are indispensable (from the left).

T(n,k) is also the number of integers m where the length of base-4 representation of m is n+k and the digit sum of the base-4 representation of 3m is 3(k+1).

			Triangle begins
   3;
   6,   6;
  10,  28,   10;
  15,  81,   81,   15;
  21, 186,  354,  186,   21;
  28, 371, 1137, 1137,  371,  28;
  36, 672, 3018, 4836, 3018, 672, 36;
  ...
There are 6 4-ary strings (10, 12, 13, 20, 23, 30) of length 2 with 1 indispensable digits and a nonzero leading digit.
There are 6 4-ary strings (11, 21, 22, 31, 32, 33) of length 2 with 2 indispensable digits and a nonzero leading digit.
There are 10 4-ary strings (111, 211, 221, 222, 311, 321, 322, 331, 332, 333) of length 3 with 3 indispensable digits and a nonzero leading digit.
Hence, T(1,0)=6, T(1,1)=6, T(2,2)=10.

J. Y. Choi, Indispensable digits for digit sums, Notes on Number Theory and Discrete Mathematics, 25(2), (2019), pp. 40-48.
J. Y. Choi, Digit sums generalizing binomial coefficients, J. Integer Seq. 22 (2019), Article 19.8.3.

Cf. A330381, A330509, A330510, A005773, A008287.

A008287(n, k) = if(n<0, 0, polcoeff((1 + x + x^2 + x^3)^n, k));
A330509(n, k) = A008287(n, 3*k-2)+A008287(n, 3*k-1) + A008287(n, 3*k);
T(n, k) = A330509(n+1,k+1) - A330509(n,k+1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jan 19 2021

T(n,k) = A330509(n+1,k+1) - A330509(n,k+1).

More terms from Michel Marcus, Jan 19 2021

cp's OEIS Frontend

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.

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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.

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A340620 T(n,k) is the number of 4-ary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit with 0 <= k <= n.

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